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_ Handle with EXTREME CARE This volume is damaged or brittle and CANNOTbe repaired! • photocopy only if necessar • return to staff do not put in bookdrop ^ary Gerstein Science Information Centre Dedekind, Richard Essays on the Theory Of Numbers QA 248 D43 c.l PASC Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/essaysintheoryofOOdedeuoft 34 -^ IN THE SAME SERIES. ON CONTINUITY AND IRRATIONAL NUMBERS, and ON THE NATURE AND MEANING OF NUMBERS. By R. Dedekind. From the German by IV. iV. Beman. Pages, 115. Cloth, 75 cents net (3s. 6d. net). GEOMETRIC EXERCISES IN PAPER-FOLDING. By T. SuNDARA Row. Edited and revised by W. W. Beman and D. E. Smith. With many half-tone engravings from pho- tographs of actual exercises, and a package of papers for folding. Pages, circa 200. Cloth, Ji.oo. net (4s. 6d. net). (In Preparation.) ON THE STUDY AND DIFFICULTIES OF MATHE- MATICS. By Augustus De Morgan. 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ESSAYS THEORY OF NUMBERS I. CONTINUITY AND IRRATIONAL NUMBERS II. THE NATURE AND MEANING OF NUMBERS RICHARD DEDEKIND AUTHORIZED TRANSLATION BY WOOSTER WOODRUFF BEMAN PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MICHIGAN CHICAGO THE OPEN COURT PUBLISHING COMPANY LONDON AGENTS Kegan Paul, Trench, Trubner & Co., Ltd. 1901 TRANSLATION COPYRIGHTED BY The Open Court Publishing Co. 1901. D^3 ELECTRONIC VERSlOM (2^q^00 vos- NO. r iT? CONTENTS I. CONTINUITY AND IRRATIONAL NUMBERS. PAGE Preface i I. Properties of Rational Numbers 3 II. Comparison of the Rational Numbers with the Points of a Straight Line 6 III. Continuity of the Straight Line 8 IV. Creation of Irrational Numbers 12 V. Continuity of the Domain of Real Numbers . . . ig VI. Operations with Real Numbers 21 VII. Infinitesimal Analysis 24 II. THE NATURE AND MEANING OF NUMBERS. Prefaces 31 I. Systems of Elements 44 II. Transformation of a System 50 III. Similarity of a Transformation. Similar Systems . . 53 IV. Transformation of a System in Itself 56 V. The Finite and Infinite 63 VI. Simply Infinite Systems. Series of Natural Numbers . 67 VII. Greater and Less Numbers 70 VIII. Finite and Infinite Parts of the Number-Series ... 81 IX. Definition of a Transformation of the Number-Series by Induction 83 X. The Class of Simply Infinite Systems 92 XI. Addition of Numbers 96 XII. Multiplication of Numbers loi XIII. Involution of Numbers 104 XIV. Number of the Elements of a Finite System .... 105 CONTINUITY AND IRRATIONAL NUMBERS CONTINUITY AND IRRATIONAL NUMBERS. A yf Y attention was first directed toward the consid- ^*^ erations which form the subject of this pam- phlet in the autumn of 1858. As professor in the Polytechnic School in Zurich I found myself for the first time obliged to lecture upon the elements of the differential calculus and felt more keenly than ever before the lack of a really scientific foundation for arithmetic. In discussing the notion of the approach of a variable magnitude to a fixed limiting value, and especially in proving the theorem that every magnitude which grows continually, but not beyond all limits, must certainly approach a limiting value, I had re- course to geometric evidences. Even now such resort to geometric intuition in a first presentation of the differential calculus, I regard as exceedingly useful, from the didactic standpoint, and indeed indispens- able, if one does not wish to lose too much time. But that this form of introduction into the differential cal- culus can make no claim to being scientific, no one will deny. For myself this feeling of dissatisfaction Was so overpowering that I made the fixed resolve to keep meditating on the question till I should find a a CONTINUITY AND purely arithmetic and perfectly rigorous foundation for the principles of infinitesimal analysis. The state- ment is so frequently made that the differential cal- culus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given ; even the most rigorous expositions of the differential cal- culus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner. Among these, for example, belongs the above-men- tioned theorem, and a more careful investigation con- vinced me that this theorem, or any one equivalent to it, can be regarded in some way as a sufficient basis for infinitesimal analysis. It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858, and a few days afterward I communicated the results of my meditations to my dear friend Dur^ge with whom I had a long and lively discussion. Later I explained these views of a scientific basis of arith- metic to a few of my pupils, and here in Braun- schweig read a paper upon the subject before the sci- entific club of professors, but I could not make up my mind to its publication, because, in the first place, the presentation did not seem altogether simple, and further, the theory itself had little promise. Never IRRATIONAL NUMBERS. 3 theless I had already half determined to select this theme as subject for this occasion, when a few days ago, March 14, by the kindness of the author, the paper Die Eleinenie der Funktionenlehre by E. Heine {serene's Journal^ Vol. 74) came into my hands and confirmed me in my decision. In the main I fully agree with the substance of this memoir, and in- deed I could hardly do otherwise, but I will frankly acknowledge that my own presentation seems to me to be simpler in form and to bring out the vital point more clearly. While writing this preface (March 20, 1872), I am just in receipt of the interesting paper Ueber die Ausdehnung eines Satzes aus der Theorie der trigonotnetrischen Reihen, by G. Cantor {Math. Annalen, Vol. 5), for which I owe the ingenious author my hearty thanks. As I find on a hasty perusal, the ax- iom given in Section II. of that paper, aside from the form of presentation, agrees with what I designate in Section III. as the essence of continuity. But what advantage will be gained by even a purely abstract definition of real numbers of a higher type, I am as yet unable to see, conceiving as I do of the domain of real numbers as complete in itself. I. PROPERTIES OF RATIONAL NUMBERS. The development of the arithmetic of rational numbers is here presupposed, but still I think it worth while to call attention to certain important 4 CONTINUITY AND matters without discussion, so as to show at the out- set the standpoint assumed in what follows. I regard the whole of arithmetic as a necessary, or at least nat- ural, consequence of the simplest arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding; the simplest act is the passing from an already-formed individual to the con- secutive new one to be formed. The chain of these numbers forms in itself an exceedingly useful instru- ment for the human mind; it presents an inexhaustible wealth of remarkable laws obtained by the introduc- tion of the four fundamental operations of arithmetic. Addition is the combination of any arbitrary repeti- tions of the above-mentioned simplest act into a sin- gle act ; from it in a similar way arises multiplication. While the performance of these two operations is al- ways possible, that of the inverse operations, subtrac- tion and division, proves to be limited. Whatever the immediate occasion may have been, whatever com- parisons or analogies with experience, or intuition, may have led thereto ; it is certainly true that just this limitation in performing the indirect operations has in each case been the real motive for a new crea- tive act ; thus negative and fractional numbers have been created by the human mind ; and in the system of all rational numbers there has been gained an in- strument of infinitely greater perfection. This system, IRRATIONAL NUMBERS. 5 which I shall denote by R^ possesses first of all a com- pleteness and self-containedness which I have desig- nated in another place* as characteristic. of a body of numbers [Zahlkorper] and which consists in this that the four fundamental operations are always perform- able with any two individuals in R, i. e., the result is always an individual of R, the single case of division by the number zero being excepted. For our immediate purpose, however, another property of the system R is still more important ; it may be expressed by saying that the system R forms. a well-arranged domain of one dimension extending to infinity on two opposite sides. What is meant by this is sufficiently indicated by my use of expressions borrowed from geometric ideas ; but just for this rea- son it will be necessary to bring out clearly the corre- sponding purely arithmetic properties in order to avoid even the appearance as if arithmetic were in need of ideas foreign to it. To express that the symbols a and b represent one and the same rational number we put a-=b 2iS well as b^a. The fact that two rational numbers «, /^ are different appears in this that the difference a — b has either a positive or negative value. In the former case a is said to be greater than ^, b less than a ; this is also indicated by the symbols a^b, b <Ca.'\ As in the latter case b — a has a positive value it follows *Vorlesungen iiber Zahlentheorie, by P. G. Lejeune Dirichlet. 2d ed. % 159. t Hence in what follows the so-called '• algebraic " greater and less are understood unless the word "absolute" is added. 6 CONTINUITY AND that b^ a, a<ib. In regard to these two ways in which two numbers may differ the followmg laws will hold: I. If a^b, and /5 > ^, then «>r. Whenever a, c are two different (or unequal) numbers, and b is greater than the one and less than the other, we shall, without hesitation because of the suggestion of geo- metric ideas, express this briefly by saying : b lies be- tween the two numbers a, c. II. If «, c are two different numbers, there are in- finitely many different numbers lying between «, c. ^ III. If a is any definite number, then all numbers of the system R fall into two classes, A\ and Ai, each of which contains infinitely many individuals ; the first class ^1 comprises all numbers a\ that are <ia^ the second class A^ comprises all numbers a^ that are >«; the number a itself may be assigned at pleasure to the first or second class, being respectively the greatest number of the first class or the least of the second. In every case the separation of the system R into the two classes A\^ A^ is such that every num- ber of the first class A\ is less than every number of the second class A%. II. COMPARISON OF THE RATIONAL NUMBERS WITH THE POINTS OF A STRAIGHT LINE. The above-mentioned properties of rational num- bers recall the corresponding relations of position of IRRATIONAL NUMBERS. 7 the points of a straight Hne L. If the two opposite directions existing upon it are distinguished by "right" and *'left," and/, q are two different points, then either / lies to the right of q, and at the same time q to the left of/, or conversely q lies to the right of/ and at the same time/ to the left of q. A third case is impossible, if p, q are actually different points. In regard to this difference in position the following laws hold : I. If / lies to the right of q, and q to the right of r, then / lies to the right of r\ and we say that q lies between the points p and r. ^ II. If /, r are two different points, then there al- ways exist infinitely many points that lie between p and r. III. If / is a definite point in Z, then all points in L fall into two classes, P\, F<i, each of which contains infinitely many individuals ; the first class Fi contains all the points /i, that lie to the left of /, and the sec- ond class P2 contains all the points /2 that lie to the right of/ ; the point / itself may be assigned at pleas- ure to the first or second class. In every case the separation of the straight line L into the two classes or portions Fi, F2, is of such a character that every point of the first class Fi lies to the left of every point of the second class F^. This analogy between rational numbers and the points of a straight line, as is well known, becomes a real correspondence when we select upon the straight 8 CONTINUITY AND line a definite origin or zero-point o and a definite unit of length for the measurement of segments. With the aid of the latter to every rational number a a cor- responding length can be constructed and if we lay this off upon the straight line to the right or left of o according as a is positive or negative, we obtain a definite end-point /, which may be regarded as the point corresponding to the number a ; to the rational number zero corresponds the point o. In this way to every rational number «, i. e., to every individual in R, corresponds one and only one point/, i. e., an in- dividual in L, To the two numbers <2, b respectively correspond the two points/, ^, and if «>^, then/ lies to the right of q. To the laws i, ii, in of the pre- vious Section correspond completely the laws i, ii, iii of the present. III. CONTINUITY OF THE STRAIGHT LINE. Of the greatest importance, however, is the fact that in the straight line L there are infinitely many points which correspond to no rational number. If the point / corresponds to the rational number a, then, as is well known, the length op is commensur- able with the invariable unit of measure used in the construction, i. e., there exists a third length, a so- called common measure, of which these two lengths are integral multiples. But the ancient Greeks already IRRATIONAL NUMBERS. g knew and had demonstrated that there are lengths in- commensurable with a given unit of length, e. g., the diagonal of the square whose side is the unitof length. If we lay off such a length from the point o upon the line we obtain an end-point which corresponds to no rational number. Since further it can be easily shown that there are infinitely many lengths which are in- commensurable with the unit of length, we may affirm: The straight line L is infinitely richer in point-indi- viduals than the domain R of rational numbers in number-individuals. If now, as is our desire, we try to follow up arith- metically all phenomena in the straight line, the do- main of rational numbers is insufficient and it becomes absolutely necessary that the instrument ^ constructed by the creation of the rational numbers be essentially improved by the creation of new numbers such that the domain of numbers shall gain the same complete- ness, or as we may say at once, the same continuity, as the straight line. The previous considerations are so familiar and well known to all that many will regard their repeti- tion quite superfluous. Still I regarded this recapitu- lation as necessary to prepare properly for the main question. For, the way in which the irrational num- bers are usually introduced is based directly upon the conception of extensive magnitudes — which itself is nowhere carefully defined — and explains number as the result of measuring such a magnitude by another lo CONTINUITY AND of the same kind.* Instead of this I demand that arithmetic shall be developed out of itself. That such comparisons with non-arithmetic no- tions have furnished the immediate occasion for the ex- tension of the number- concept may, in a general way, be granted (though this was certainly not the case in the introduction of complex numbers); but this surely is no sufficient ground for introducing these foreign notions into arithmetic, the science of numbers. Just as negative and fractional rational numbers are formed by a new creation, and as the laws of operating with these numbers must and can be reduced to the laws of operating with positive integers, so we must en- deavor completely to define irrational numbers by means of the rational numbers alone. The question only remains how to do this. The above comparison of the domain R of rational numbers with a straight line has led to the recognition of the existence of gaps, of a certain incompleteness or discontinuity of the former, while we ascribe to the straight line completeness, absence of gaps, or con- tinuity. In what then does this continuity consist? Everything must depend on the answer to this ques- tion, and only through it shall we obtain a scientific basis for the investigation of all continuous domains. By vague remarks upon the unbroken connection in ♦The apparent advantage of the generality of this definition of number disappears as soon as we consider complex numbers. According to my view, on the other hand, the notion of the ratio between two numbers of the same kind can be clearly developed only after the introduction of irrational num- bers. IRRATIONAL NUMBERS. ii the smallest parts obviously nothing is gained ; the problem is to indicate a precise characteristic of con- tinuity that can serve as the basis for valid deductions. For a long time I pondered over this in vain, but finally I found what I was seeking. This discovery will, perhaps, be differently estimated by different people ; the majority may find its substance very com- monplace. It consists of the following. In the pre- ceding section attention was called" to the fact that every point p of the straight line produces a separa- tion of the same into two portions such that every point of one portion lies to the left of every point of the other. I find the essence of continuity in the con- verse, i. e., in the following principle : ** If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this di- vision of all points into two classes, this severing of the straight line into two portions." As already said I think I shall not err in assuming that every one will at once grant the truth of this statement ; the majority of my readers will be very much disappointed in learning that by this common- place remark the secret of continuity is to be revealed. To this I may say that I am glad if every one finds the above principle so obvious and so in harmony with his own ideas of a line ; for I am utterly unable to adduce any proof of its correctness, nor has any 12 CONTINUITY AND one the power. The assumption of this property of the line is nothing else than an axiom by which we attribute to the line its continuity, by which we find continuity in the line. If space has at all a real ex- istence it is not necessary for it to be continuous ; many of its properties would remain the same even were it discontinuous. And if we knew for certain that space was discontinuous there would be nothing to prevent us, in case we so desired, from filling up its gaps, in thought, and thus making it continuous ; this filling up would consist in a creation of new point- individuals and would have to be effected in accord- ance with the above principle. IV. CREATION OF IRRATIONAL NUMBERS. From the last remarks it is sufficiently obvious how the discontinuous domain R of rational numbers may be rendered complete so as to form a continuous domain. In Section I it was pointed out that every rational number a effects a separation of the system R into two classes such that every number a\ of the first class A\ is less than every number a<i of the second class A^ ; the number a is either the greatest number of the class A\ or the least number of the class ^j. If now any separation of the system R into two classes A\^ Ai, is given which possesses only this characteris- tic property that every number a\ in A\ is less than every number a^ in Ai, then for brevity we shall call IRRATIONAL NUMBERS. 13 such a separation a cut [Schnitt] and designate it by (^1, ^2). We can then say that every rational num- ber a produces one cut or, strictly speaking, two cuts, which, however, we shall not look upon as essentially different ; this cut possesses, besides, the property that either among the numbers of the first class there ex- ists a greatest or among the numbers of the second class a least number. And conversely, if a cut pos- sesses this property, then it is produced by this great- est or least rational number. But it is easy to show that there exist infinitely many cuts not produced by rational numbers. The following example suggests itself most readily. Let Z> be a positive integer but not the square of an integer, then there exists a positive integer A. such that If we assign to the second class ^2? every positive rational number ^2 whose square is ;> Z>, to the first class A\ all other rational numbers a\, this separation forms a cut {A\, A2), i. e., every number ai is less than every number a^. For if ^i = 0, or is negative, then on that ground ai is less than any number a2, because, by definition, this last is positive ; if ai is positive, then is its square <Z>, and hence ai is less than any positive number a^ whose square is >>Z>. But this cut is produced by no rational number. To demonstrate this it must be shown first of all that there exists no rational number whose square z^V. M CONTINUITY AND Although this is known from the first elements of the theory of numbers, still the following indirect proof may find place here. If there exist a rational number whose square =Z>, then there exist two positive in- tegers /, «, that satisfy the equation and we may assume that u is the least positive integer possessing the property that its square, by multipli- cation by D, may be converted into the square of an integer /. Since evidently X«</<(A+1)«, the number u* ■=t — \u is a positive integer certainly less than u. If further we put f = Du—\i, f is likewise a positive integer, and we have ^ ^JSy*^ which is contrary to the assumption respecting u. Hence the square of every rational number x is either <^D or >i?. From this it easily follows that there is neither in the class Ai a greatest, nor in the class A^ a least number. For if we put we have and __ _2x{I) — x^) (3^:2 + Z))«* IRRATIONAL NUMBERS. 15 If in this we assume ^ to be a positive number from the class A\, then x^ <^D, and hence jk>^ and y^ <CD. Therefore J^' likewise belongs to the class Ai. But if we assume ^ to be a number from the class A^, then x^^D, and hence y <,x, ^>0, and y^^D. Therefore y likewise belongs to the class A^. This cut is therefore produced by no rational number. In this property that not all cuts are produced by rational numbers consists the incompleteness or dis- continuity of the domain R of all rational numbers. Whenever, then, we have to do with a cut (^A\, A^) produced by no rational number, we create a new, an irratio7ial number a, which we regard as completely defined by this cut {A\, A2); we shall say that the number a corresponds to this cut, or that it produces this cut. From now on, therefore, to every definite cut there corresponds a definite rational or irrational number, and we regard two numbers as different or unequal always and only when they correspond to es- sentially different cuts. In order to obtain a basis for the orderly arrange- ment of all real J i. e., of all rational and irrational numbers we must investigate the relation between any two cuts {A\, Ai) and (j5i, ^2) produced by any two numbers a and j8. Obviously a cut (^1, Ai) is given completely when one of the two classes, e. g., the first A\ is known, because the second A^ consists of all rational numbers not contained in A\, and the characteristic property of such a first class lies in this i6 CONTINUITY AND that if the number a\ is contained in it, it also con- tains all numbers less than ax. If now we compare two such first classes A\^ B\ with each other, it may happen 1. That they are perfectly identical, i. e., that every number contained in A\ is also contained in B\y and that every number contained in B\ is also contained in A\. In this case A% is necessarily identical with B'iy and the two cuts are perfectly identical, which we denote in symbols by a = ^ or ^ = a. But if the two classes A\^ B\ are not identical, then there exists in the one, e. g., in A\y a number a'\ = b\ not contained in the other B\ and conse- quently found in Bi ; hence all numbers b\ contained in B\ are certainly less than this number a\=^b'i and therefore all numbers bi are contained in A\. 2. If now this number a'\ is the only one in A\ that is not contained in B\, then is every other number a\ contained in A\ also contained in B\ and is conse- quently <a'i, i. e., a\ is the greatest among all the numbers a\^ hence the cut (^i, y^a) is produced by the rational number a=^a'\^^b\. Concerning the other cut (^i, B'l) we know already that all numbers b\ in B\ are also contained in A\ and are less than the number a!\=^b*i which is contained in Bi ; every other number b% contained in B^ must, however, be greater than ^'a, for otherwise it would be less than tf'i, therefore contained in Ax and hence in Bx ; hence b\ is the least among all numbers contained in B%^ IRRATIONAL NUMBERS, 17 and consequently the cut (^1, Bi) is produced by the same rational number ^z=b\=^ a\ = a. The two cuts are then only unessentially different. 3. If, however, there exist in Ax at least two differ- ent numbers a'\^=b\ and a'\^=b"i, which are not con- tained in B\, then there exist infinitely many of them, because all the infinitel)^ many numbers lying between a\ and a'\ are obviously contained in A\ (Section I, II) but not in B\. In this case we say that the num- bers a and /8 corresponding to these two essentially different cuts (^1, Ai) and (^1, ^2) are different, and further that a is greater than ^, that ^ is less than a, which we express in symbols by a >> /? as well as ^ < a. It is to be noticed that this definition coincides com- pletely with the one given earlier, when a, )8 are ra- tional. The remaining possible cases are these : 4. If there exists in B\ one and only one number b'i=^a'2, that is not contained in Ai then the two cuts (^1, A2) and {B\, Bi) are only unessentially different and they are produced by one and the same rational number a = ^'2 = b\ = /?. 5. But if there are in B\ at least two numbers which are not contained in A\, then )8>a, a</8. As this exhausts the possible cases, it follows that of two different numbers one is necessarily the greater, the other the less, which gives two possibilities. A third case is impossible. This was indeed involved in the use of the comparative (greater, less) to desig- i8 CONTINUITY AND nate the relation between a, fi ; but this usq has only now been justified. In just such investigations one needs to exercise the greatest care so that even with the best intention to be honest he shall not, through a hasty choice of expressions borrowed from other no- tions already developed, allow himself to be led into the use of inadmissible transfers from one domain to the other. If now we consider again somewhat carefully the case a>)8 it is obvious that the less number j8, if rational, certainly belongs to the class A\ \ for since there is in A\ a number a'\=^b'<i which belongs to the class B<i, it follows that the number )8, whether the greatest number in B\ or the least in B^ is certainly <tf'i and hence contained in A\. Likewise it is ob- vious from a> j8 that the greater number a, if rational, certainly belongs to the class Bi, because a> «'i. Com- bining these two considerations we get the following result : If a cut is produced by the number a then any rational number belongs to the class A\ or to the class Ai according as it is less or greater than a; if the number a is itself rational it may belong to either class. From this we obtain finally the following : If a> )8, i. e., if there are infinitely many numbers in A\ not contained in B\ then there are infinitely many such numbers that at the same time are different from a and from j8; every such rational number ^ is <Ca, because IRRATIONAL NUMBERS. 19 it is contained in A\ and at the same time it is >/3 because contained in B'^. V. CONTINUITY OF THE DOMAIN OF REAL NUMBERS. In consequence of the distinctions just established the system H of all real numbers forms a well-arranged domain of one dimension ; this is to mean merely that the following laws prevail : I. If a>^, and y8!>y, then is also a>y. We shall scy that the number )8 lies between a and y. II. If a, y are any two different numbers, then there exist infinitely many different numbers ^ lying between a, y. III. If a is any definite number then all numbers of the system H fall into two classes Hi and 2^2 each of which contains infinitely many individuals; the first class 2li comprises all the numbers ai that are less than a, the second 2X2 comprises all the numbers a.1 that are greater than a ; the number a itself may be assigned at pleasure to the first class or to the second, and it is respectively the greatest of the first or the least of the second class. In each case the separation of the system H into the two classes 1X\, H2 is such that every number of the first class 2(i is smaller than every number of the second class 2(2 and we say that this separation is produced by the number a. For brevity and in order not to weary the reader I suppress the proofs of these theorems which follow ao CONTINUITY AND immediately from the definitions of the previous sec- tion. Beside these properties, however, the domain 2? possesses also continuity \ i. e., the following theorem is true : IV. If the system H of all real numbers breaks up into two classes 2li, ^(2 such that every number ai of the class 1X\ is less than every number ai of the class ^2 then there exists one and only one number a by which this separation is produced. Proof. By the separation or the cut of H into 2Ii and 2(2 we obtain at the same time a cut (y^i, Ai) of the system R of all rational numbers which is de- fined by this that A\ contains all rational numbers of the class 2(i and A^ all other rational numbers, i. e., all rational numbers of the class 2(2. Let a be the perfectly definite number which produces this cut (^1, Ai). If ^ is any number different from a, there are always infinitely many rational numbers c lying between a and yS. If ^<a, then ^<a; hence ^r be- longs to the class A\ and consequently also to the class 2(1, and since at the same time fi<ic then fi also belongs to the same class 2ti, because every number in 2(2 is greater than every number c in 2li. But if /3>a, then is ^>a; hence c belongs to the class A'^ and consequently also to the class ^(2, and since at the same time )8>r, then ^ also belongs to the same class 2(2, because every number in 2li is less than every number c in IXi- Hence every number /8 differ- IRRATIONAL NUMBERS. 21 ent from a belongs to the class id or to the class ils according as )8<a or ^>a; consequently a itself is either the greatest number in 2(i or the least number in ^2, i- e., a is one and obviously the only number by which the separation of R into the classes Zli, 2(2 is produced. Which was to be proved. VI. OPERATIONS WITH REAL NUMBERS. To reduce any operation with two real numbers a, fi to operations with rational numbers, it is only necessary from the cuts {A\, Ai), {Bi, B2) produced by the numbers a and /3 in the system R to define the cut (Ci, C2) which is to correspond to the result of the operation, y. I confine myself here to the discus- sion of the simplest case, that of addition. If c is any rational number, we put it into the class Ci, provided there are two numbers one ai in ^1 and one /fi in Bi such that their sum ^i-f-^i>^; all other rational numbers shall be put into the class C^. This separation of all rational numbers into the two classes Ci, C2 evidently forms a cut, since every number ci in Ci is less than every number ^2 in C2. If both a and /Q are rational, then every number ci contained in Ci is <a+j8, because ^i<a, h<P, and therefore «i + ^i <a-{- ft; further, if there were contained in C2 a num- ber C2<a-\- /3, hence a-\- /3 = C2 +/, where / is a pos- itive rational number, then we should have .2-(a-J/) + (i8-i/), 22 CONTINUITY AND \ which contradicts the definition of the number c% be- cause a — \p is a number in A\, and ^ — \p a number in B\ ; consequently every number ci contained in Cg is >a-|-^. Therefore in this case the cut (Ci, Ci) is produced by the sum a + )8. Thus we shall not violate the definition which holds in the arithmetic of rational numbers if in all cases we understand by the sum a-\- fi oi any two real numbers a, /? that number y by which the cut (Ci, Ci) is produced. Further, if only one of the two numbers a, ^ is rational, e. g., o, it is easy to see that it makes no difference with the sum y = a-|-/^ whether the number a is put into the class A\ or into the class A^. Just as addition is defined, so can the other ope- rations of the so-called elementary arithmetic be de- fined, viz., the formation of differences, products, quotients, powers, roots, logarithms, and in this way we arrive at real proofs of theorems (as, e. g., |/2-l/3 ^1/6), which to the best of my knowledge have never been established before. The excessive length that is to be feared in the definitions of the more complicated operations is partly inherent in the nature of the subject but can for the most part be avoided. Very useful in this connection is the notion of an interval, i. e., a system A of rational numbers possessing the follow- ing characteristic property: if a and a' are numbers of the system A, then are all rational numbers lying between a and a' contained in A. The system R of all rational numbers, and also the two classes of any IRRATIONAL NUMBERS. 23 cut are intervals. If there exist a rational number a\ which is less and a rational number a^ which is greater than every number of the interval A, then A is called a finite interval ; there then exist infinitely many num- bers in the same condition as a\ and infinitely many in the same condition as a^ ; the whole domain R breaks up into three parts Ax, A, A^ and there enter two per- fectly definite rational or irrational numbers ai, 02 which may be called respectively the lower and upper (or the less and greater) limits of the interval; the lower limit ai is determined by the cut for which the system A\ forms the first class and the upper ai by the cut for which the system A^ forms the second class. Of every rational or irrational number a lying between ai and 02 it may be said that it lies within the interval A. If all numbers of an interval A are also numbers of an interval B, then A is called a portion of B. Still lengthier considerations seem to loom up when we attempt to adapt the numerous theorems of the arithmetic of rational numbers (as, e. g., the theo- rem {a-\-b')c=^ac-\-bc') \.o any real numbers. This, however, is not the case. It is easy to see that it all reduces to showing that the arithmetic operations possess a certain continuity. What I mean by this statement may be expressed in the form of a general theorem : '* If the number \ is the result of an operation per- formed on the numbers a, /?, y, . . . and A. lies within the interval Z, then intervals A, B, C, . . . can be 24 CONTINUITY AND taken within which lie the numbers a, /3, y, . . , such that the result of the same operation in which the numbers a, /3, y, . . . are replaced by arbitrary num- bers of the intervals A, B, C, . . , is always a number lying within the interval Z." The forbidding clumsi- ness, however, which marks the statement of such a theorem convinces us that something must be brought in as an aid to expression ; this is, in fact, attained in the most satisfactory way by introducing the ideas of variable magnitudes^ functions^ limiting values, and it would be best to base the definitions of even the sim- plest arithmetic operations upon these ideas, a matter which, however, cannot be carried further here. VII. INFINITESIMAL ANALYSIS. Here at the close we ought to explain the connec- tion between the preceding investigations and certain fundamental theorems of infinitesimal analysis. We say that a variable magnitude x which passes through successive definite numerical values ap- proaches a fixed limiting value a when in the course of the process x lies finally between two numbers be- tween which a itself lies, or, what amounts to the same, when the difference x — a taken absolutely be- comes finally less than any given value different from zero. One of the most important theorems may be stated in the following manner: *'If a magnitude .jc grows IRRATIONAL NUMBERS. 25 continually but not beyond all limits it approaches a limiting value." I prove it in the following way. By hypothesis there exists one and hence there exist infinitely many numbers a^ such that x remains continually •<a2; I designate by 2I2 the system of all these numbers 02, by 2(1 the system of all other numbers ai ; each of the latter possesses the propdbty that in the course of the process x becomes finally >ai, hence every number ai is less than every number ai and consequently there exists a number a which is either the greatest in 2ti or the least in 2(2 (V, iv). The former cannot be the case since x never ceases to grow, hence a is the least number in 2I2 Whatever number ai be taken we shall have finally ai<<:j:<a, i. e., x approaches the limiting value a. This theorem is equivalent to the principle of con- tinuity, i. e., it loses its validity as soon as we assume a single real number not to be contained in the do- main K ; or otherwise expressed : if this theorem is correct, then is also theorem iv. in V. correct. Another theorem of infinitesimal analysis, likewise equivalent to this, which is still oftener employed, may be stated as follows : **If in the variation of a magnitude x we can for every given positive magni- tude S assign a corresponding position from and after which X changes by less than S then x approaches a limiting value." This converse of the easily demonstrated theorem 26 CONTINUITY AND that every variable magnitude which approaches a limiting value finally changes by less than any given positive magnitude can be derived as well from the preceding theorem as directly from the principle of continuity. I take the latter course. Let 8 be any positive magnitude (i. e., S>0), then by hypothesis a time will come after which x will change by less than 8, i. e., if at this time^^: has the value «, then afterwards we shall continually have x"^ a — 8 and X <C^a-\-h. I now for a moment lay aside the original hypothesis and make use only of the theorem just demonstrated that all later values of the variable x lie between two assignable finite values. Upon this I base a double separation of all real numbers. To the sys- tem 2t2 I assign a number 02 (e. g., «-|- 8) when in the course of the process x becomes finally <a2 ; to the system 2(i I assign every number not contained in 2I2; if ai is such a number, then, however far the process may have advanced, it will still happen infinitely many times that a:>a2. Since every number ai is less than every number a2 there exists a perfectly definite num- ber a which produces this cut (2(i, 2t2) of the system H and which I will call the upper limit of the variable x which always remains finite. Likewise as a result of the behavior of the variable x a second cut (3i, 32) of the system ^ is produced ; a number ^2 (e.g., a — 8) is assigned to BV when in the course of the pro- cess j: becomes finally >/?; every other number ^82, to be assigned to 32» has the property that x is never IKRA TIONAL NUMBERS. 27 finaliy > )82 ; therefore infinitely many times x becomes <;)82 ; the number fi by which this cut is produced I call the lower limiting value of the variable x. The two numbers a, yS are obviously characterised by the following property: if c is an arbitrarily small positive magnitude then we have always finally a: < a -f e and ^>i8 — e, but never finally ^<a — c and never finally a:>)8-|-€. Now two cases are possible. If a and /8 are different from each other, then necessarily a>^, since continually a2>A; the variable x oscillates, and, however far the process advances, always under- goes changes whose amount surpasses the value (a — ^) — 2c where e is an arbitrarily small positive magnitude. The original hypothesis to which I now return contradicts this consequence ; there remains only the second case a = ^ and since it has already been shown that, however small be the positive magni- tude e, we always have finally x <Ca-}- e and x > )8 — e, X approaches the limiting value a, which was to be proved. These examples may suffice to bring out the con- nection between the principle of continuity and in- finitesimal analysis. THE NATURE AND MEANING OF NUMBERS PREFACE TO THE FIRST EDITION. TN science nothing capable of proof ought to be ac- ^ cepted without proof. Though this demand seems so reasonable yet I cannot regard it as having been met even in the most recent methods of laying the foundations of the simplest science; viz., that part of logic which deals with the theory of numbers.* In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number- concept entirely independent of the notions or intui- tions of space and time, that I consider it an imme- diate result from the laws of thought. My answer to the problems propounded in the title of this paper is, then, briefly this : numbers are free creations of the human mind ; they serve as a means of apprehending more easily and more sharply the difference of things. It is only through the purely logical process of build- ing up the science of numbers and by thus acquiring *Of the works which have come under my observation I mention the val- uable Lehrbuch der Arithmetik und Algebra of E. Schroder (Leipzig, 1873), which contains a bibliography of the subject, and in addition the memoirs of Kronecker and von Helmholtz upon the Number-Concept and upon Counting and Measuring (in the collection of philosophical essays published in honor of E. Zeller, Leipzig, 1887). The appearance of these memoirs has induced me to publish my own views, in many respects similar but in foundation essentially different, which I formulated many years ago in absolute inde- pendence of the works of others. 32 THE NA TURE AND the continuous number-domain that we are prepared accurately to investigate our notions of space and time by bringing them into relation with this number- domain created in our mind.* If we scrutinise closel}' what is done in counting an aggregate or number of things, we are led to consider the ability of the mind to relate things to things, to let a thing corre- spond to a thing, or to represent a thing by a thing, an ability without which no thinking is possible. Upon this unique and therefore absolutely indispen- sable foundation, as I have already affirmed in an an- nouncement of this paper,t must, in my judgment, the whole science of numbers be established. The design of such a presentation I had formed before the publication of my paper on Continuity, but only after its appearance and with many interruptions occa- sioned by increased official duties and other necessary labors, was I able in the years 1872 to 1878 to commit to paper a first rough draft which several mathemati- cians examined and partially discussed with me. It bears the same title and contains, though not arranged in the best order, all the essential fundamental ideas of my present paper, in which they are more carefully elaborated. As such main points I mention here the sharp distinction between finite and infinite (64), the notion of the number \_Anzahl'\ of things (161), the ♦See Section III. of my memoir, Continuity and Irrational Numbers (Braunschweig, 1872), translated at pages 8 et seq. of the present volume, + Dirichlet's Vorlesungen iiber Zahlentheorie, third edition, 1879, § 163, note on page 470. MEANING OF NUMBERS. 33 proof that the form of argument known as complete induction (or the inference from n\.on-\-\^\s really conclusive (59), (60), (80), and that therefore the definition by induction (or recursion) is determinate and consistent (126). This memoir can be understood by any one pos- sessing what is usually called good common sense ; no technical philosophic, or mathematical, knowledge is in the least degree required. But I feel conscious that many a reader will scarcely recognise in the shadowy forms which I bring before him his numbers which all his life long have accompanied him as faith- ful and familiar friends ; he will be frightened by the long series of simple inferences corresponding to our step-by-step understanding, by the matter-of-fact dis- section of the chains of reasoning on which the laws of numbers depend, and will become impatient at being compelled to follow out proofs for truths which to his supposed inner consciousness seem at once evi- dent and certain. On the contrary in just this possi- bility of reducing such truths to others more simple, no matter how long and apparently artificial the series of inferences, I recognise a convincing proof that their possession or belief in them is never given by inner consciousness but is always gained only by a more or less complete repetition of the individual inferences. I like to compare this action of thought, so difficult to trace on account of the rapidity of its performance, with the action which an accomplished reader per- 34 THE NA TURE AND forms in reading ; this reading always remains a more or less complete repetition of the individual steps which the beginner has to take in his wearisome spelling-out ; a very small part of the same, and there- fore a very small effort or exertion of the mind, is suffi- cient for the practised reader to recognise the correct, true word, only with very great probability, to be sure) for, as is well known, it occasionally happens that even the most practised proof-reader allows a typographical error to escape him, i. e., reads falsely, a thing which would be impossible if the chain of thoughts associated with spelling were fully repeated. So from the time of birth, continually and in increas- ing measure we are led to relate things to things and thus to use that faculty of the mind on which the creation of numbers depends ; by this practice con- tinually occurring, though without definite purpose, in our earliest years and by the attending formation of judgments and chains of reasoning we acquire a store of real arithmetic truths to which our first teach- ers later refer as to something simple, self-evident, given in the inner consciousness ; and so it happens that many very complicated notions (as for example that of the number \Anzahl'\ of things) are errone- ously regarded as simple. In this sense which I wish to express by the word formed after a well-known saying aCi 6 dvOpioiro^ apiOfx-qril^u, I hope that the follow- ing pages, as an attempt to establish the science of numbers upon a uniform foundation will find a gener- MEANING OF NUMBERS, 35 ous welcome and that other mathematicians will be led to reduce the long series of inferences to more moderate and attractive proportions. In accordance with the purpose of this memoir I restrict myself to the consideration of the series of so-called natural numbers. In what way the gradual extension of the number-concept, the creation of zero, negative, fractional, irrational and complex numbers are to be accomplished by reduction to the earlier notions and that without any introduction of foreign conceptions (such as that of measurable mag- nitudes, which according to my view can attain per- fect clearness only through the science of numbers), this I have shown at least for irrational numbers in my former memoir on Continuity (1872); in a way wholly similar, as I have already shown in Section III. of that memoir,* may the other extensions be treated, and I propose sometime to present this whole subject in systematic form. From just this point of view it appears as something self-evident and not new that every theorem of algebra and higher analysis, no mat- ter how remote, can be expressed as a theorem about natural numbers, — a declaration I have heard repeat- edly from the lips of Dirichlet. But I see nothing meritorious — and this was just as far from Dirichlet's thought — in actually performing this wearisome cir- cumlocution and insisting on the use and recognition of no other than rational numbers. On the contrary, ♦Pages 8 et seq. of the present volume. 36 THE NATURE AND the greatest and most fruitful advances in mathematics and other sciences have invariably been made by the creation and introduction of new concepts, rendered necessary by the frequent recurrence of complex phe- nomena which could be controlled by the old notions only with difficulty. On this subject I gave a lecture before the philosophic faculty in the summer of 1854 on the occasion of my admission as privat-docent in Gottingen. The scope of this lecture met with the approval of Gauss ; but this is not the place to go into further detail. Instead of this I will use the opportunity to make some remarks relating td my earlier work, mentioned above, on Continuity and Irrational Numbers. The theory of irrational numbers there presented, wrought out in the fall of 1853, is based on the phenomenon (Section IV,)* occurring in the domain of rational numbers which I designate by the term cut \_Schnitt'\ and which I was the first to investigate carefully ; it culminates in the proof of the continuity of the new domain of real numbers (Section V., iv. ).f It appears to me to be somewhat simpler, I might say easier, than the two theories, different from it and from each other, which have been proposed by Weierstrass and G. Cantor, and which likewise are perfectly rigorous. It has since been adopted without essential modifica- tion by U.' Dini in his Fondamenti per la teorica delle ♦Pages 12 et seq. of the present volume. f Page 2o of the present volume. MEANING OF NUMBERS. 37 funzioni di variabili reali (Pisa, 1878); but the fact that in the course of this exposition my name happens to be mentioned, not in the description of the purely arithmetic phenomenon of the cut but when the au- thor discusses the existence of a measurable quantity corresponding to the cut, might easily lead to the sup- position that my theory rests upon the consideration of such quantities. Nothing could be further from the truth; rather have I in Section III.* of my paper advanced several reasons why I wholly reject the in- troduction of measurable quantities ; indeed, at the end of the paper I have pointed out with respect to their existence that for a great part of the science of space the continuity of its configurations is not even a necessary condition, quite aside from the fact that in works on geometry arithmetic is only casually men- tioned by name but is never clearly defined and there- fore cannot be employed in demonstrations. To ex- plain this matter more clearly I note the following example : If we select three non-collinear points A, B, C at pleasure, with the single limitation that the ratios of the distances AB^ AC, BC are algebraic numbers,t and regard as existing in space only those points M, for which the ratios of AM, BM, CM to AB are likewise algebraic numbers, then is the space made up of the points M, as is easy to see, everywhere dis- * Pages 8 et seq. of the present volume. tDirlchlet's Vorlesungen iiber Zahlentheorie, § 159 of the second edition, § 160 of the third. 38 THE NATURE AND continuous; but in spite of this discontinuity, and de- spite the existence of gaps in this space, all construc- tions that occur in Euclid's EiementSy can, so far as I can see, be just as accurately effected as in perfectly continuous space ; the discontinuity of this space would not be noticed in Euclid's science, would not be felt at all. If any one should say that we cannot conceive of space as anything else than continuous, I should venture to doubt it and to call attention to the fact that a far advanced, refined scientific training is demanded in order to perceive clearly the essence of continuity and to comprehend that besides rational quantitative relations, also irrational, and besides al- gebraic, also transcendental quantitative relations are conceivable. All the more beautiful it appears to me that without any notion of measurable quantities and simply by a finite system of simple thought-steps man can advance to the creation of the pure continuous number-domain ; and only by this means in my view is it possible for him to render the notion of continu- ous space clear and definite. The same theory of irrational numbers founded upon the phenomenon of the cut is set forth in the Introduction d la thdorie des fonctions d^une variable by J. Tannery (Paris, 1886). If I rightly understand a passage in the preface to this work, the author has thought out his theory independently, that is, at a time when not only my paper, but Dini's Fondamenti mentioned in the same preface, was unknown to him. MEANING OF NUMBERS. 39 This agreement seems to me a gratifying proof that my conception conforms to the nature of the case, a fact recognised by other mathematicians, e. g., by Pasch in his Einleitung in die Differential- und Integral- rechnung (Leipzig, 1883). But I cannot quite agree with Tannery when he calls this theory the develop- ment of an idea due to J. Bertrand and contained in his Traiti d^arithmitique, consisting in this that an ir- rational number is defined by the specification of all rational numbers that are less and all those that are greater than the number to be defined. As regards this statement which is repeated by Stolz — apparently without careful investigation — in the preface to the second part of his Vorlesungen iiber allgemeine Arith- metik (Leipzig, 1886), I venture to remark the follow- ing : That an irrational number is to be considered as fully defined by the specification just described, this conviction certainly long before the time of Ber- trand was the common property of all mathematicians who concerned themselves with the notion of the irrational. [Just this manner of determining it is ir the mind o! every computer wb ^ '— ' rational root of an equ*^ as Bertrand does exc edition, of the year the irrational ni^ able quantities, ^ already set fort^ celebrated defii 40 MEANING OF NUMBERS. ity of two ratios (^Elements, V., 5). J This same most ancient conviction has been the source of my theory as well as that of Bertrand and many other more or less complete attempts to lay the foundations for the introduction of irrational numbers into arithmetic. But though one is so far in perfect agreement with Tannery, yet in an actual examination he cannot fail to observe that Bertrand's presentation, in which the phenomenon of the cut in its logical purity is not even mentioned, has no similarity whatever to mine, inasmuch as it resorts at once to the existence of a measurable quantity, a notion which for reasons men- tioned above I wholly reject. Aside from this fact this method of presentation seems also in the succeed- ing definitions and proofs, which are based on the postulate of this existence, to present gaps so essential that I still regard the statement made in my paper (Section VI. ), * that the theorem V^' V^ 3 = l/6 has no- where yet been strictly demonstrated, as justified with respect to this work also, so excellent in many other " ■'^^'^^ -^riH "nth which I was unacquainted at that j-fc^ariia^^-.. R. Deoekind. s PREFACE TO THE SECOND EDITION. 'T^HE present memoir soon after its appearance met -■- with both favorable and unfavorable criticisms ;. indeed serious faults were charged against it. I have been unable to convince myself of the justice of these charges, and I now issue a new edition of the memoir, which for some time has been out of print, without change, adding only the following notes to the first preface. The property which I have employed as the deii nition of the infinite system had been pointed out be- fore the appearance of my paper by G. Cantor (^Ein Beitrag zur Mannigfaltigkeitslehre, CrelWs Journal, Vol. 84, 1878), as also by Bolzano {Paradoxien des Unend- lichen, § 20, 1851). But neither of these authors made the attempt to use this property for the definition of the infinite and upon this foundation to establish with rigorous logic the science of numbers, an4 just in this consists the content of my wearisome labor which in all its essentials I had completed several years before the appearance of Cantor's memoir and at a time when the work of Bolzano was unknown to me even by name. For the benefit of those who are interested in and understand the difficulties of such an investi- 42 THE NATURE AND gation, I add the following remark. We can lay down an entirely different definition of the finite and infinite, which appears still simpler since the notion of sim- ilarity of transformation is not even assumed, viz. : **A system S is said to be finite when it may be so transformed in itself (36) that no proper part (6) of 6" is transformed in itself; in the contrary case S is called an infinite system." Now let us attempt to erect our edifice upon this new foundation! We shall soon meet with serious difficulties, and I believe myself warranted in saying that the proof of the perfect agreement *of this defini- tion with the former can be obtained only (and then easily) when we are permitted to assume the series of natural numbers as already developed and to make use of the final considerations in (131); and yet noth- ing is said of all these things in either the one defini- tion or the other! From this we can see how very great is the number of steps in thought needed for such a remodeling of a definition. About a year after the publication of my memoir I became acquainted with G. Frege's Grundlagen der Arithmetikf which had already appeared in the year 1884. However different the view of the essence of number adopted in that work is from my own, yet it contains, particularly from § 79 on, points of very close contact with my paper, especially with my defi- nition (44). The agreement, to be sure, is not easy to discover on account of the different form of expres- MEANING OF NUMBERS. 43 sion ; but the positiveness with which the author speaks of the logical inference from nton-\-l (page 93, below) shows plainly that here he standi upon the same ground with me. In the meantime E. Schroder's Vorlesungen ilber die Algebra der Logik has been almost completed (1890-1891). Upon the importance of this extremely suggestive work, to which I pay my highest tribute, it is impossible here to enter further ; I will simply confess that in spite of the remark made on p. 253 of Part I., I have retained my somewhat clumsy symbols (8) and (17); they make no claim to be adopted generally but are intended simply to serve the purpose of this arithmetic paper to which in my view they are better adapted than sum and product symbols. R. Dedekind. Harzburg, August 24, 1893. THE NATURE AND MEANING OF NUMBERS. SYSTEMS OF ELEMENTS. 1. In what follows I understand by thing every object of our thought. In order to be able easily to speak of things, we designate them by symbols, e. g., by letters, and we venture to speak briefly of the thing a ox oi a simply, when we mean the thing de- noted by a and not at all the letter a itself. A thing is completely determined by all that can be affirmed or thought concerning it. A thing a is the same as b (identical with /^), and b the same as a^ when all that can be thought concerning a can also be thought con- cerning b, and when all that is true of b can also be thought of a. That a and b are only symbols or names for one and the same thing is indicated by the nota- tion a--=b, and also hy b = a. If further /5 = r, that is, if c as well as « is a symbol for the thing denoted by b, then is also a=^c. If the above coincidence of the thing denoted by a with the thing denoted by b does not exist, then are the things a, b said to be dif- ferent, a is another thing than b, <^ "another thing than MEANING OF NUMBERS. 45 a\ there is some property belonging to the one that does not belong to the other. 2. It very frequently happens that different things, «, b, c, . . . for some reason can be considered from a common point of view, can be associated in the mind, and we say that they form a system S; we call the things a, b, c, . . . elements of the system S, they are contained in S\ conversely, S consists of these elements. Such a system S (an aggregate, a mani- fold, a totality) as an object of our thought is like- wise a thing (1); it is completely determined when with respect to every thing it is determined whether it is an element of .S" or not.* The system S is hence the same as the system T, in symbols S^T, when every element of ^ is also element of T, and every element of T is also element of S. For uniformity of expression it is advantageous to include also the spe- cial case where a system S consists of a single (one and only one) element a, i. e., the thing a is element of S, but every thing different from a is not an ele- ment of S. On the other hand, we intend here for certain reasons wholly to exclude the empty system which contains no element at all, although for other * In what manner this determination is brought about, and whether we know a way of deciding upon it, is a matter of indifference for all that follows; the general laws to be developed in no way depend upon it; they hold under all circumstances. I mention this expressly because Kronecker not long ago [Crelle's Journal, Vol. 99, pp. 334-336^ has endeavored to impose certain limi- tations upon the free formation of concepts in mathematics which I do not believe to be justified ; but there seems to be no call to enter upon this mat- ter with more detail until the distinguished mathematician shall have pub- lished his reasons for the necessity or merely the expediency of these limi- tations. 46 THE NATURE AND investigations it may be appropriate to imagine such a system. 3. Definition. A system A is said to be part of a system S when every element of A is also element of S. Since this relation between a system A and a sys- tem S will occur continually in what follows, we shall express it briefly by the symbol A^S. The inverse symbol S^ A, by which the same fact might be ex- pressed, for simplicity and clearness I shall wholly avoid, but for lack of a better word I shall sometimes say that 5 is whole oi A^ by which I mean to express that among the elements of S are found all the ele- ments of A. Since further every element j of a system S by (2) can be itself regarded as a system, we can hereafter employ the notation s^S. 4. Theorem. A^A, by reason of (3). 5. Theorem. If ^ 3^ and ^3^, then ^=^. The proof follows from (3), (2). 6. Definition. A system A is said to be a proper [echter'] part of S, when A is part of S, but different from S. According to (5) then S is not a part of A, i. e., there is in 6* an element which is not an element of ^. 7. Theorem. If AiB and B^C, which may be denoted briefly by A^B^C, then is A^C, and A is certainly a proper part of C, if ^ is a proper part of -5 or if ^ is a proper part of C. The proof follows from (3), (6). 8. Definition. By the system compounded out of MEANING OF NUMBERS. 47 any systems A, B, C, . . . to be denoted by ITt {A, B^ C, . . .) we mean that system whose elements are de- termined by the following prescription : a thing is considered as element of HT {A^ B^ C, . . .) when and only when it is element of some one of the systems ^, ^, C, . . ., i. e., when it is element of Aj or B, or C, . . . We include also the case where only a single system A exists; then obviously HT {A) = A. We observe further that the system 2\l (^A, B, C, . . .) compounded out of A, B, C, . . . is carefully to be dis- tinguished from the system whose elements are the systems A, B, C, . . . themselves. 9. Theorem, The systems A, B, C, . . . are parts oim(A,B, C, . . .)• The proof follows from (8), (3). 10. Theorem. U A, B, C, . . . are parts of a sys- tem S, then is Vd (A, B, C, . . .) ^ S. The proof follows from (8), (3). 11. Theorem. If B is part of one of the systems A, B, C, . . . then is B^VTi {A, B, C, . . .). The proof follows from (9), (7). 12. Theorem. If each of the systems T', Q, . . . is part of one of the systems A, B, C, . . . then is HI {F, (2, . . •) ^VTi {A, B, C, . . .). The proof follows from (11), (10). 13. Theorem. If A is compounded out of any of the systems F, Q, . . . then is A'^VTi (F, Q, . . .)• Proof. For every element of A is by (8) element of one of the systems F, Q, . . ., consequently by (8) 48 THE NATURE AND also element of ^ (Z', Q, . . .), whence the theorem follows by (3). 14. Theorem. If each of the systems A, B, C, . , , is compounded out of any of the systems F^ Q, . . . then is m^A, B, c, . . .)nn{F, (2, • • o The proof follows from (13), (10). 15. Theorem. If each of the systems P, Q, . . . is part of one of the systems A, B^ C, . . ., and if each of the latter is compounded out of any of the former, then is m{p, (2, . . .)=m{A, B, c, . . .). The proof follows from (12), (14), (5). 16. Theorem. If A = m (B, Q) and B = m (Q, B) then is m {A, B) = m (B, B). Proof. For by the preceding theorem (15) m (^, B) as well as VCi (B, B)--=m {B, Q, B), 17. Definition. A thing g is said to be conmion element of the systems A^ B, C, . . ., \i it is contained in each of these systems (that is in A and in B and in C . . .). Likewise a system 7" is said to be a co7n- mon part of A, B, C, . . . when Z'is part of each of these systems; and hy the community \Gemeinheit'\ of the systems A, B, C, . . . we understand the perfectly determinate system ® {A, B, C, . . .') which consists of all the common elements g oi A, B, C, . . . and MEANING OF NUMBERS. 49 hence is likewise a common part of those systems. We again include the case where only a single system A occurs; then ^\A) (is to be put) r=A. But the case may also occur that the systems A, B, C, . . . possess no common element at all, therefore no com- mon part, no community ; they are then called sys- tems without common part, and the symbol ^ (^, B, C, . . .) is meaningless (compare the end of (2)). We shall however almost always in theorems con- cerning communities leave it to the reader to add in thought the condition of their existence and to dis- cover the proper interpretation of these theorems for the case of non-existence. 18. Theorem. Every common part of A, B^ C, . . . is part of ® {A, B, C, . . .)• The proof follows from (17). 19. Theorem. Every part of ^ {A, B, C, . . .) is common part of ^, ^, C, . . . The proof follows from (17), (7). 20. Theorem. If each of the systems A, B, C, . . .^ is whole (3) of one of the systems P, Q, . . . then is ® (Z', (2, . . .) ^ ® (^» ^, C, . , .) Proof. For every element of @ (F, Q, . . .) is common element oi F, Q, . . ., therefore also common element of ^4, ^, C, . . ., which was to be proved. 50 THE NA TURE AND II. TRANSFORMATION OF A SYSTEM. 21. Definition.* By a transformation \Abbildung'\ <^ of a system S we understand a law according to which to every determinate element s of 5 there be- longs a determinate thing which is called the trans- form of s and denoted by <^(^); we say also that <^(i-) corresponds to the element s^ that <^{s) results or is produced from s by the transformation </>, that s is transformed mto cf>{s) by the transformation <^. If now 7" is any part of S, then in. the transformation <}> oi S is likewise contained a determinate transformation of T, which for the sake of simplicity may be denoted by the same symbol <f> and consists in this that to every element / of the system T there corresponds the same transform <^(/), which / possesses as element of .S; at the same time the system consisting of all transforms </> (/) shall be called the transform of T and be denoted by <f>{T), by which also the significance of <t>(S) is defined. As an example of a transformation of a sys- tem we may regard the mere assignment of deter- minate symbols or names to its elements. The sim- plest transformation of a system is that by which each of its elements is transformed into itself ; it will be called the t'dentica/ transiormaition of the system. For convenience, in t!^e following theorems (22), (23), (24), which deal with an arbitrary transformation <^ of ♦See Dirichlet's VorUsungen Ubcr Zahlentheorte, 3d edition, 1879, § 163. MEANING OF NUMBERS. 51 an arbitrary system S, we shall denote the transforms of elements s and parts T respectively by / and Z"; in addition we agree that small and capital italics without accent shall always signify elements and parts of this system S. 22. Theorem.* If ^3^, then ^' ^ i5'. Proof. For every element of A' is the transform of an element contained in A, and therefore also in B. and is therefore element of B\ which was to be proved. 23. Theorem. The transform of ITt (^, B, C, . . .) is m {A\ B\ C, . . .). Proof. If we denote the system 2TT {A, B, C, . . .) which by (10) is likewise part of 6* by M, then is every element of its transform M' the transform ?n' of an element m of M; since therefore by (8) m is also ele- ment of one of the systems A, B, C, . . . and conse- quently m' element of one of the systems A', B\ C, . . ., and hence by (8) also element of Vii (A', B\ C, . . .), we have by (3) M'^m{A\B\ C\ . . .). On the other hand, since A, B, C, . . . are by (9) parts of M, and hence A', B\ C, . . . by (22) parts of M', we have by (10) m{A', B', C, . . .)^M'. By combination with the above we have by (5) the theorem to be proved M'=m{A\B', r, . . .)• ♦ See theorem 27. 52 THE NATURE AND 24. Theorem.* The transform of every common part of A J B, C, . . .y and therefore that of the com- munity (5 {A, B, C, . . .) is part of © (^', B', C\ . . .). Proof. For by (22) it is common part of A', B\ C, . . ., whence the theorem follows by (18). 25. Definition and theorem. If <^ is a transforma- tion of a system S, and ^ a transformation of the transform S' = <f>(S'), there always results a transfor- mation of S, compounded^ out of <^ and «/^, which con- sists of this that to every element s oi S there corres- ponds the transform where again we have put <f>(^s)=s\ This transforma- tion can be denoted briefly by the symbol {f/.<f> or il/<f>, the transform 0(s) by »/'<^(i') where the order of ihe symbols <f>, ij/ is to be considered, since in general the symbol <j>\j/ has no interpretation and actually has meaning only when if/{s')^s. If now x signifies a transformation of the system il/{s') = if/<f,(s) and rj the transformation x'A ^^ the system ^S" compounded out of «A and x» then is x^(0 = X'A('f') =='7(0 = '7«^ W; therefore the compound transformations x^ and ri<f> coincide for every element s of ^S", i. e., x^^=V^- ^^ accordance with the meaning of and y this theorem can finally be expressed in the form X'^<f>-=\^-4>» ♦See theorem 29. tA confusion of this compounding of transformations with that of sys- tems of elements is hardly to be feared. MEANING OF NUMBERS. 53 and this transformation compounded out of <^, i//, y^ can be denoted briefly by x'A*^* III. SIMILARITY OF A TRANSFORMATION. SIMILAR SYSTEMS. 26. Definition. A transformation <^ of a system S is said to be similar [ahnlich'] or distinct, when to dif- ferent elements a, b of the system S there always cor- respond different transforms «'^<^(«), b' ^=<^{l)). Since in this case conversely from /=r/' we always have sz=it, then is every element of the system S' = <f> (S) the transform / of a single, perfectly determi- nate element s of the system S, and we can therefore set over against the transformation <^ of 6* an inverse transformation of the system 6", to be denoted by ^, which consists in this that to every element / of S' there corresponds the transform $(/)=rj-, and obvi- ously this transformation is also similar. It is clear that ^(S') = Sf that further <^ is the inverse transformation belonging to 5 and that the transformation ^cf> com- pounded out of <l> and $ by (25) is the identical trans- formation of S (21). At once we have the following additions to II., retaining the notation there given. 27. Theorem.* If A' 3^', then A^B. Proof. For if a is an element of A then is a' an element of A\ therefore also of B', hence ^=:b', where b is an element of B ; but since from a' = b' we always * See theorem 22. 54 THE NATURE AND have a = by then is every element oi A also element of B, which was to be proved. 28. Theorem. If ^' = ^', then ^ =^. The proof follows from (27), (4), (5). 29. Theorem.* If G = ^{A, B, C, . . .), then G' = ^{A\ B\ C, . . .). Proof. Every element of ^{A', B' , C, . . .) is certainly contained in S\ and is therefore the trans- form ^' of an element ^ contained in .S; but since ^' is common element of A', B', C, . . . then by (27) must g be common element of Aj B, C, . . . therefore also element of G', hence every element of ©(^', B' , C\ . . .) is transform of an element ^ of G, therefore element of G\ i. e., ^(^', B\ C\ . . .)3 6^', and ac- cordingly our theorem follows from (24), (5). 30. Theorem. The identical transformation of a system is always a similar transformation. 31. Theorem. If <^ is a similar transformation of .Sand j/f a similar transformation of <I>{S), then is the transformation {(/<{> of S, compounded of <f> and \f/, a sim- ilar transformation, and the associated inverse trans- formation xjr^^^ij/. Proqf. For to different elements a, h of 6* corre- spond different transforms a' = <f>{a), l>' =z<f){lf), and to these again different transforms ij/{a') = if/<f>(^a), if/(^d'") = \}/<f>(^d) and therefore {l/<f> is a. similar transfor- mation. Besides every element i/r<^(j-):=,/r(/) of the system il/<f>{S) is transformed by if into s' = <f>(s) and * See theorem 24. MEANING OF NUMBERS. 55 this by $ into j, therefore y\i^{s) is transformed by '^^ into i", which was to be proved. 32. Definition. The systems R, S are .said to be similar when there exists such a similar transforma- tion ^ oi S that <^(^)=^, and therefore ^(7?) = ^. Obviously by (30) every system is similar to itself. 33. Theorem. If R, S are similar systems, then every system Q similar to R is also similar to S. Proof. For if <J[), y\i are similar transformations of S, R such that f\>{S)=R, ij/(R)=Q, then by (31) il/<f> is a similar transformation of S such that il/<f>(S) = Q, which was to be proved. 34. Definition. We can therefore separate all sys- tems into classes by putting into a determinate class all systems Q, R, S, . . ., and only those, that are similar to a determinate system R, the representative of the class ; according to (33) the class is not changed by taking as representative any other system belong- ing to it. 35. Theorem. If ^, 6" are similar systems, then is every part of 6" also similar to a part of R, every proper part of S also similar to a proper part of R. Proof. For if <^ is a similar transformation of 5, <I>(^S) = R, and T^S, then by (22) is the system sim- ilar to T<I>{T)^R; if further T is proper part of 6", and s an element of S not contained in T, then by (27) the element <l>{s) contained in R cannot be contained in <j>{T); hence <fi{T) is proper part of R, which was to be proved. 56 THE NATURE AND IV. TRANSFORMATION OF A SYSTEM IN ITSELF. 36. Definition. If <^ is a similar or dissimilar trans- formation of a system S, and <^(»S) part of a system Z, then <^ is said to be a transformation of 5 in Z, and we say S is transformed by <^ in Z. Hence we call ^ a transformation of the system S in itself, when <^(5)3»S', and we propose in this paragraph to investi- gate the general laws of such a transformation <^. In doing this we shall use the same notations as in II. and again put <^(i-) = /, <f>(^T)= T\ These trans- forms s\ T' are by (22), (7) themselves again ele- ments or parts of g) like all things designated by italic letters. 37. Definition. K \s called a chain \Kette\, when K'^K. We remark expressly that this name does not in itself belong to the part K of the system S, but is given only with respect to the particular transfor- mation <^ ; with reference to another transformation of the system S in itself K can very well not be a chain. 38. Theorem. ^ is a chain. 39. Theorem. The transform K* of a chain ^is a chain. Proof. For from K"^ K it follows by (22) that {^K')'^K\ which was to be proved. 40. Theorem. If A is part of a chain K, then is also ^'^ A-. MEANING OF NUMBERS. 57 Proof. For from A^ K it follows by (22) that A'^K\ and since by (37) K"^K, therefore by (7) A'^K, which was to be proved. 41. Theorem. If the transform A' is part of a chain Z, then is there a chain K, which satisfies the conditions A^K, K'iL; and VCi{A, L) is just such a chain K. Proof. If we actually put X=2Tt {A, L), then by (9) the one condition A^K is fulfilled. Since further by (23) K' = m{.A\ L') and by hypothesis A'^L, L'^Lf then by (10) is the other condition K'^L also fulfilled and hence it follows because by (9) L^K, that also K'^K, i. e. , ^ is a chain, which was to be proved. ',^ x^ I 42. Theorem. A system J/" compounded simply out of chains ^4, ^, C, . . . is a chain. Proof. Since by (23) M' = VCi{A\ B', C, . . .) and by hypothesis ^'3 j9, B'iB, C'iC, . . . therefore by (12) M'^M, which was to be proved. 43. Theorem. The community G of chains A ^, C, ... is a chain. Proof. Since by (17) G is common part of A, B, C, . . ., therefore by (22) G' common part of A\ B\ C\ . . ., and by hypothesis A'^A, B'^B, C?>C, . . ., then by (7) G' is also common part of A, B, C, . . . and hence by (18) also part of G, which was to be proved. 44. Definition. If A is any part of S, we will de- note by A^ the community of all those chains (e.g.^ S) 58 THE NATURE AND of which A is part ; this community A^ exists (17) be- cause A is itself common part of all these chains. Since further by (43) A^ is a chain, we will call A^ the chain of the system A, or briefly the chain of A. This definition too is strictly related to the fundamen- tal determinate transformation <^ of the system .S in itself, and if later, for the sake of clearness, it is necessary we shall at pleasure use the symbol <i>o(^) instead of A^, and likewise designate the chain of A corresponding to another transformation <o by iiio{A). For this very important notion the following theorems hold true. 45. Theorem. A^A^. Proof. For A is common part of all those chains whose community is A„, whence the theorem follows by (18). 46. Theorem. {A.y^A,. Proof. For by (44) A„ is a chain (37). 47. Theorem. If A is part of a chain iT, then is also A, ^X. Proof. For A^ is the community and hence also a common part of all the chains K, of which A is part. 48. Remark. One can easily convince himself that the notion of the chain A„ defined in (44) is com- pletely characterised by the preceding theorems, (45), (46), (47). 49. Theorem. A'i^A^y. The proof follows from (45), (22). MEANING OF NUMBERS. 59 50. Theorem. A'^A^. The proof follows from (49), (46), (7). 51. Theorem. If ^ is a chain, then A^=A. Proof. Since A is part of the chain A, then by (47) A J A, whence the theorem follows by (45), (5). 52. Theorem. If ^3^, then ^3 ^„. The proof follows from (45), (7). 53. Theorem. If BiA„, then B^^A^, and con- versely. Proof. Because A„ is a chain, then by (47) from B^A„, we also get B^^A^; conversely, ii B^^A^, then by (7) we also get B^A^, because by (45) B^B^. 54. Theorem. If B^A, then is B,^A^. The proof follows from (52), (53). 55. Theorem. If BiA„, then is also B'iA^. Proof. For by (53) BJA„ and since by (50) B'^B^, the theorem to be proved follows by (7). The same result, as is easily seen, can be obtained from (22), (46), (7), or also from (40). 56. Theorem. If BiA„, then is {B^y^{A,y. The proof follows from (53), (22). 57. Theorem and definition. (A^)' = {A')g, i. e., the transform of the chain of A is at the same time the chain of the transform of A. Hence we can desig- nate this system in short by A'^ and at pleasure call it the chain-transfor7ti or transform- chain of A. With the clearer notation given in (44) the theorem might be expressed by <^(<^,(^)) = <^,(<^(^)). Proof. If for brevity we put {A')^ = L^ Z is a <5o THE NATURE AND chain (44) and by (45) A'^L\ hence by (41) there ex- ists a chain ^satisfying the conditions A^K, K'^L\ hence from (47) we have A^^K, therefore {A^'^K\ and hence by (7) also {A^'^L, i. e., (^,)'n^')o. Since further by (49) A'^(^A^\ and by (44), (39) (^o)' is a chain, then by (47) also (^')»^(^o)', whence the theorem follows by combining with the preceding result (5). 58. Theorem. A^^Xtl^A, A'^), i. e., the chain of A is compounded out of A and the transform-chain of ^. Proof. If for brevity we again put Z=A', = {A,y = {A'\ ^nd X=m<iA, Z), then by (45) A'iL, and since Z is a chain, by (41) the same thing is true of ZT; since further ^3 Z" (9), therefore by (47) On the other hand, since by (45) AiA^, and by (46) also LiA^y then by (10) also K^A,, whence the theorem to be proved AqZ=K follows by combining with the preceding result (5). 59. Theorem of complete induction. In order to show that the chain A^ is part of any system 2 — be this latter part of S or not — it is sufficient to show, p. that A^% and MEANING OF NUMBERS. 6i 0-. that the transform of every common element of Aq and 5 is likewise element of 5. Proof. For if p is true, then by (45) the com- munity 6^==^(^„, 5) certainly exists, and by (18) A^G\ since besides by (17) then is G also part of our system S, which by <^ is transformed in itself and at once by (55) we have also G'^Aq. If then o- is likewise true, i. e., if G*^%, then must G' as common part of the systems A^y % by (18) be part of their community G, i. e., (? is a chain (37), and since, as above noted, A i G, then by (47) is also AJG, and therefore by combination with the preceding re- sult G = A^, hence by (17) also A^^X which was to be proved. 60. The preceding theorem, as will be shown later, forms the scientific basis for the form of demonstra- tion known by the name of complete induction (the inference from n to n-{-l) ; it can also be stated in the following manner : In order to show that all ele- ments of the chain A^ possess a certain property (£ (or that a theorem 5 dealing with an undetermined thing n actually holds good for all elements n of the chain A^) it is sufficient to show p. that all elements a of the system A possess the property (£ (or that 5 holds for all ^'s) and 0-. that to the transform n' of every such element n of Ao possessing the property (£, belongs the same 62 THE NATURE AND property (£ (or that the theorem 5, as soon as it holds for an element « of A^^ certainly must also hold for its transform «'). Indeed, if we denote by 2 the system of all things possessing the property (£ (or for which the theorem S holds) the complete agreement of the present man- ner of stating the theorem with that employed in (59) is immediately obvious. 61. Theorem. The chain of HI (^, -^, C, . . .) is ZR(^o, ^o, Q, • . .)• Proof. If we designate by M the former, by K the latter system, then by (42) ^ is a chain. Since then by (45) each of the systems A, JB, C, , . . is part of one of the systems A^, B^y C^, . . ., and therefore by (12) M^K, then by (47) we also have On the other hand, since by (9) each of the systems Ay Bj C, . . . is part of J/, and hence by (45), (7) also part of the chain M^, then by (47) must also each of the systems A^, B^, C^, ... be part of J/^) therefore by (10) whence by combination with the preceding result fol- lows the theorem to be proved M^^K (5). 62. Theorem. The chain of ^{A, By C, . . .) is part of ^(^„ B„ C,y . . .). Proof. If we designate by G the former, by K the latter system, then by (43) X is a chain. Since then each of the systems A^, B^, Q, . . . by (45) is whole MEAlSriNG OF NUMBERS. 63 of one of the systems A, B, C, . . ., and hence by (20) G^K, therefore by (47) we obtain the theorem to be proved G^^K. 63. Theorem. If K'iL^K, and therefore X is a chain, L is also a chain. If the same is proper part of Kj and U the system of all those elements of K which are not contained in Z, and if further the chain Uq is proper part of K, and V the system of all those elements of K which are not contained in U^, then is K=m{^U„ F) andZ = irr(^'o, V). If finally Z=ir' then F3 V. The proof of this theorem of which (as of the two preceding) we shall make no use may be left for the reader. V. THE FINITE AND INFINITE. 64. Definition.* A system ^ is said to he injimte when it is similar to a proper part of itself (32); in the contrary case S is said to be a jfintte system. 65. Theorem. Every system consisting of a single element is finite. Proof. For such a system possesses no proper part (2), (6). * If one does not care to employ the notion of similar systems (32) he must say: 5 is said to be infinite, when there is a proper part of 5" (6) in which 5 can be distinctly (similarly) transformed (26), (36). In this form I submitted the definition of the infinite which forms the core of my whole investigation in September, 1882, to G. Cantor and several years earlier to Schwarz and Weber. All other attempts that have come to my knowledge to distinguish the infinite from the finite seem to me to have met with so little success that 1 think I may be permitted to forego any criticism of them. 64 THE NATURE AND 66. Theorem. There exist infinite systems. Proof.* My own realm of thoughts, i. e., the to- tality 5 of all things, which can be objects of my thought, is infinite. For if s signifies an element of Sj then is the thought /, that s can be object of my thought, itself an element of S. If we regard this as transform <^ {/) of the element s then has the transfor- mation <^ of S^ thus determined, the property that the transform S' is part of S\ and S' is certainly proper part of S, because there are elements in S (e. g., my own ego) which are different from such thought / and therefore are not contained in S' . Finally it is clear that if ^, b are different elements of S, their trans- forms a\ b' are also different, that therefore the trans- formation ^ is a distinct (similar) transformation (26). Hence S is infinite, which was to be proved. 67. Theorem. If R, S are similar systems, then is R finite or infinite according as S is finite or infinite. Proof. If S is infinite, therefore similar to a proper part S' of itself, then if R and S are similar, S' by (33) must be similar to R and by (35) likewise similar to a proper part of R^ which therefore by (33) is itself similar to R\ therefore R is infinite, which was to be proved. 68. Theorem. Every system S, which possesses an infinite part is likewise infinite ; or, in other words, every part of a finite system is finite. ♦A similar consideration is found in § 13 of the Paradoxien des Unend' lichen by Bolzano (Leipzig, 185X). MEANING OF NUMBERS. 65 Proof. If T is infinite and there is hence such a similar transformation \^ of T, that ^(T') is a proper part of T, then, if T is part of ^, we can extend this transformation i/^ to a transformation <^ of ^ in which, if x denotes any element oi S, we put i^{s')^=\\i{s') or <^(j)=r=j- according as ^ is element of T'or not. This transformation <^ is a similar one ; for, if a, b denote different elements of S, then if both are contained in T, the transform (f> (a) =: \f/ (a) is different from the transform <f> (^/?) = k}/ {b) , because i/^ is a similar transfor- mation ; if further a is contained in T, but l> not, then is <fi(^a)=il/{a) different from cfi{b')==3, because {{/{a) is contained in T; if finally neither a nor b is con- tained in J" then also is <^(^) = ^ different from <^(<^)=/^, which was to be shown. Since further il/(T) is part of T, because by (7) also part of 6*, it is clear that also <l){S)iS. Since finally iJ/^T) is proper part of T' there exists in 7' and therefore also in S, an element /, not contained in 1/^(7") = <^(7') ; since then the transform ^(i-) of every element s not contained in T'is equal to s, and hence is different from /, / cannot be contained in<^(^); hence <f>{S) is proper part of ^ and conse- quently S is infinite, which was to be proved. 69. Theorem. Every system which is similar to a part of a finite system, is itself finite. The proof follows from (67), (68). 70. Theorem. If a is an element of S, and if the aggregate T of all the elements of S different from a is finite, then is also ^S finite. 66 THE NATURE AND Proof. We have by (6-1) to show that if <^ denotes any similar transformation of .S in itself, the trans- form </)(-S') or S' is never a proper part of 6" but al- ways = 6". Obviously 6'=:2n(^, T) and hence by (23), if the transforms are again denoted by accents, 5' = 2n(«', ^'), and, on account of the similarity of the transformation <^, a is not contained in T' (26). Since further by hypothesis S'^S^ then must a! and like- wise every element of T' either =d!, or be element of T. If then — a case which we will treat first — a is not contained in T\ then must T'^Tand hence T = T, because ^ is a similar transformation and because T'is a finite system; and since a, as remarked, is not con- tained in T' , i.e., not in T, then must a' ^=a, and hence in this case we actually have S' = S 2,s was stated. In the opposite case when a is contained in T' and hence is the transform b' of an element b contained in T, we will denote by U the aggregate of all those elements u of r, which are different from b; then T=Vri{b,U) and by (15) S=m {a, b, U), hence S' .= m {a', a, U'). We now determine a new transformation j/r of T' in which we put \p{b) = a'y and generally \l/{ii)^u\ whence by (23) xp{T) = 'm{a', U'). Obviously xj/ is a similar transformation, because <j> was such, and be- cause a is not contained in £/"and therefore also a' not in U'. Since further a and every element u is differ- ent from b then (on account of the similarity of <^) must also a' and every element u' be different from a and consequently contained, in T] hence \f/(T)^T MEANING OF NUMBERS, 67 and since T is finite, therefore must j/'CT") =7", and "m^a, U')=T. From this by (15) we obtain m{a\ a, C/') = m{a, T) • i. e., according to the preceding S' = S. Therefore in this case also the proof demanded has been se- cured. VI. SIMPLY INFINITE SYSTEMS. SERIES OF NATURAL NUMBERS. 71. Definition. A system N is said to be simply infinite when there exists a similar transformation <^ of N\Vi itself such that iV appears as chain (44) of an element not contained in <^(A'). We call this ele- ment, which we shall denote in what follows by the symbol 1, the base-element of N and say the simply infinite system N is set in order \_geordnet'\ by this transformation <^. If we retain the earlier convenient symbols for transforms and chains (IV) then the es- sence of a simply infinite system N consists in the existence of a transformation ^ of iVand an element 1 which satisfy the following conditions a, y8, y, 8 : a. N'^N. y. The element 1 is not contained in iV'. 8. The transformation <^ is similar. Obviously it follows from a, y, 8 that every simply in- finite system N'ls actually an infinite system (64) be- cause it is similar to a proper part N' of itself. 68 THE NA TURK AND 72. Theorem. In every infinite system S a. simply infinite system JV \s contained as a part. Proof. By (64) there exists a similar transforma- tion <f> of ^S'such that <f>{S) or S' is a proper part of »S'; hence there exists an element 1 in ^ which is not contained in .5". The chain JV=1^, which corresponds to this transformation <^ of the system S in itself (44), is a simply infinite system set in order by <^ ; for the characteristic conditions a, ft, y, 8 in (71) are obvi- ously all fulfilled. 73. Definition. If in the consideration of a simply infinite system iV set in order by a transformation <^ we entirely neglect the special character of the ele- ments^ simply retaining their distinguishability and taking into account only the relations to one another in which they are placed by the order- setting trans- formation <{>, then are these elements called natural, numbers or ordinal numbers or simply numbers, and the base-element 1 is called the base-number of the number- series N. With reference to this freeing the elements from every other content (abstraction) we are justified in calling numbers a free creation of the human mind. The relations or laws which are derived entirely from the conditions a, ft, y, 8 in (71) and therefore are al- ways the same in all ordered simply infinite systems, whatever names may happen to be given to the indi- vidual elements (compare 134), form the first object of the science of nu?nbers or arithmetic. From the general notions and theorems of IV. about the transformation MEANING OF NUMBERS. 6g of a system in itself we obtain immediately the follow- ing fundamental laws where a, b, . . . iti, n, . . . always denote elements of N, therefore numbers, A\ B, C, . . . parts of Nj a', b' , . . . m', n' , . . . A', B\ C . . . the corresponding transforms, which are produced by the order-setting transformation ^ and are always ele- ments or parts of N; the transform n' of a number n is also called the number fo Hawing n. 74. Theorem. Every number n by (45) is con- tained in its chain n^ and by (53) the condition n^m^ is equivalent to nj^in^. 75. Theorem. By (57) «',= («,)':= («'),. 76. Theorem. By (46) n'.'^n,. 77. Theorem. By (58) n,=^lXi{n, n\). 78. Theorem. iV=:>n(l, ^'), hence every num- ber different from the base-number 1 is element of iV', i. e., transform of a number. The proof follows from (77) and (71). 79. Theorem. iV^is the only number-chain con- taining the base-number 1. Proof. For if 1 is element of a number-chain K^ then by (47) the associated chain N^K, hence N=K, because it is self-evident that K^N. 80. Theorem of complete induction (inference from n to n). In order to show that a theorem holds for all numbers « of a chain m^, it is sufficient to show, p. that it holds for n = m, and 0-. that from the validi4:y of the theorem for a num- 70 THE NATURE AND ber n of the chain m„ its validity for the following number «' always follows. This results immediately from the more general theorem (59) or (60). The most frequently occurring case is where w = 1 and therefore ni„ is the complete number-series iV. VII. GREATER AND LESS NUMBERS. 81. Theorem. Every number « is different from the following number ti! . Proof by complete induction (80) : p. The theorem is true for the number « = 1, be- cause it is not contained in N' (71), while the follow- ing number 1' as transform of the number 1 contained in iVis element of N\ <T. If the theorem is true for a number n and we put the following number n' =p, then is n different from /, whence by (26) on account of the similarity (71) of the order-setting transformation <^ it follows that n', and therefore /, is different from /'. Hence the theorem holds also for the number/ following «, which was to be proved. 82. Theorem. In the transform-chain n\ of a num- ber n by (74), (75) is contained its transform n', but not the number n itself. Proof by complete induction (80) : p. The theorem is true for n='i, because 1\ = JV^, MEANING OF NUMBERS. 71 and because by (71) the base-number 1 is not con- tained in N' . <r. If the theorem is true for a number n, and we again put n =p, then is n not contained in /^, there- fore is it different from every number g contained in /^, whence by reason of the similarity of </> it follows that n', and therefore/, is different from every num- ber / contained in p'„, and is hence not contained in /^. Therefore the theorem holds also for the number / following n, which was to be proved. 83. Theorem. The transform-chain n\ is proper part of the chain n„. The proof follows from (76), (74), (82). 84. Theorem. From m^ = n^ it follows that m^=n. Proof. Since by (74) m is contained in m^, and w^=«^^in(«, «^) by (77), then if the theorem were false and hence m different from n, m would be contained in the chain n\, hence by (74) also m^^n\, i. e., n^^n\; but this contradicts theorem (83). Hence our theorem is es- tablished. 85. Theorem. If the number n is not contained in the number-chain K, then is K^n\. Proof by complete induction (80) : p. By (78) the theorem is true for n^\. (T. If the theorem is true for a number n, then is it also true for the following number / = ?/; for if / is not contained in the number-chain J^, then by (40) n also cannot be contained in A and hence by our 72 THE NA TURK AND hypothesis K^n\\ now since by (77) n\=p^=z V(l (a p'o)i hence K^VTi (/, p'o) and / is not contained in Ky then must K^p\, which was to be proved. 86. Theorem. If the number n is not contained in the number-chain K, but its transform n is, then K=n\. Proof. Since n is not contained in K^ then by (85) K^n\y and since n'^K, then by (47) is also n\^Kf aiid hence Kz=n\, which was to be proved. 87o Theorem. In every number-chain K there ex- ists one, and by (84) only one, number k^ whose chain K = K. Proof. If the base-number 1 is contained in Kj then by (79) K:=N^=1„. In the opposite case let Z be the system of all numbers not contained in Ki since the base-number 1 is contained in Z, but Z is only a proper part of the number-series N, then by (79) Z cannot be a chain, i. e. , Z* cannot be part of Z] hence there exists in Z a number «, whose trans- form n' is not contained in Z, and is therefore certainly contained in K'^ since further n is contained in Z, and therefore not in K, then by (86) ^=«'^, and hence k = n'y which was to be proved. 88. Theorem. If w, n are different numbers then by (83), (84) one and only one of the chains m^, n„ is proper part of the other and either n„^m\ or mj^n'„. Proof. If n is contained in w^, and hence by (74) also n„im^, then m can not be contained in the chain ;;„ (because otherwise by (74) we should have rn^^n^, MEANING OF NUMBERS. 73 therefore m„=^n„, and hence by (84) also m^:in) and thence it follows by (85) that nj^m\. In the contrary case, when n is not contained in the chain w^, we must have by (85) m^^n\, which was to be proved. 89. Definition. The number m is said to be less than the number n and at the same time n greater than m, in symbols when the condition is fulfilled, which by (74) may also be expressed n 3 m\. 90. Theorem. If m, n are any numbers, then al- ways one and only one of the following cases A, jx, v occurs : A. m = n, n = mj i. e., ?n„=zn^ fji. irK^n, n^niy i. e., n^im\ V. m^ n, n<Cm, \. e., m„-^n\. Proof. For if \ occurs (84) then can neither y, nor V occur because by (83) we never have n^in\. But if X does not occur then by (88) one and only one of the cases /x, v occurs, which was to be proved, 91. Theorem. «<;/. Proof. For the condition for the case v in (90) is fulfilled by m^=n' . 92. Definition. To express that m is either =« or <«, hence not >« (90) we use the symbols mKn ox also n^m 74 THE NATURE AND and we say tn is at viost equal to «, and n is at least equal to m. 93. Theorem. Each of the conditions m<n, m<^n, n^^m^ is equivalent to each of the others. Proof. For if mKn, then from X, fx in (90) we always have n„?>iti„, because by (7G) m\^m. Con- versely, if n^^m„, and therefore by (74) also nim^, it fol- lows from m^ = VCi{jn, m\) that either n = m, or n^in\, _ i. e., n^m. Hence the condition m<7i is equivalent to n.ivi^. Besides it follows from (22), (27), (75) that this condition n^^m„ is agaiij equivalent to n\im\, . i. e., by /t in (90) to m<^n\ which was to be proved. 94. Theorem. Each of the conditions is equivalent to each of the others. The proof follows immediately from (93), if we replace in it m by 7n', and from yu, in (90). - 95. Theorem. If l<^in and m^n or if l^m, and m<^n^ then is I <Cn. But if /<w and m<ny then is l<n. Proof. For from the corresponding conditions (89), (93) m,il\ and n,im,, we have by (7) n,^l\ and the same thing comes also from the conditions m^^l^ and n„^m\, because in consequence of the former we have also m\M\. Finally from m„M^ and n^^m^ we have also «,^/^, which was to be proved. 96. Theorem. In every part 7" of iV there exists one and only one least number /', i. e., a number k MEANING OF NUMBERS. 75 which is less than every other number contained in T. If 7" consists of a single number, then is it also the least number in T. Proof. Since T„ is a chain (44), then by (87) there exists one number k whose chain k^=^ T^, Since from this it follows by (45), (77) that T^VTi^k, k\), then first must k itself be contained in T (because other- wise T^k\, hence by (47) also T,^k\, i. e., k^k'^, which by (83) is impossible), and besides every num- ber of the system 7", different from k, must be con- tained in k\, i. e., be >>^ (89), whence at once from (90) it follows that there exists in T one and pnly one least number, which was to be proved. 97. Theorem. The least number of the chain n^ is Hj and the base-number 1 is the least of all numbers. Proof. For by (74), (93) the condition m^n^ is equivalent to m^n. Or our theorem also follows im- mediately from the proof of the preceding theorem, because if in that we assume T=n^, evidently /^=^« (51). 98. Definition. If n is any number, then will we denote by Z„ the system of all numbers that are not greater than n, and hence not contained in n\. The condition by (92), (93) is obviously equivalent to each of the following conditions : mKn, m<^n', n^^m^. 99. Theorem. 1^Z„ and «^Z„. 76 THE NATURE AND The proof follows from (98) or from (71) and (82). 100. Theorem. Each of the conditions equivalent by (98) m^Z^, w<«, fn<in'y nj^m^ is also equivalent to the condition Proof. For if m^Z^., and hence m^n, and if l^Z^y and hence /</«, then by (95) also IKn, i. e., /3Z„; if therefore tn^Z„y then is every element /of the system Z^ also element of Z^, i. e. , Z^ 3 Z„. Conversely, if Z„, 3 Z„, then by (7) must also m 3 Z„, because by (99) m^Z^y which was to be proved. 101. Theorem. The conditions for the cases X, /m, V in (90) may also be put in the following form : \. m = n, n = m, Z„^^=Z,^ fx. m<.n, n^m, Z^^Z,, V. m^fiy n<,my Z„,iZ^. The proof follows immediately from (90) if we ob- serve that by (100) the conditions n^^m^ and ZJiZ^ are equivalent. 102. Theorem. Zi = l. Proof. For by (99) the base-number 1 is con- tained in Z\y while by (78) every number different from 1 is contained in 1'^, hence by (98) not in Zi, which was to be proved. 103. Theorem. By (98) i\^=2Tt(Z,, n\). 104. Theorem. n=^^{Z^y n^), i. e., n is the only common element of the system Z„ and n^. Proof. From (99) and (74) it follows that n is • MEANING OF NUMBERS. 77 contained in Z„ and n^ \ but every element of the chain n„ different from n by (77) is contained in n\^ and hence by (98) not in Z^, which was to be proved/ 105. Theorem. By (91), (98) the number tC is not contained in Z„. 106. Theorem. If nK^tiy then is Z^ proper part of Z^ and conversely. Proof. If m<^n, then by (100) Z^^Z„, and since the number n, by (99) contained in Z„, can by (98) not be contained in Z^ because n^m, therefore Z^ is proper part of Z„. Conversely if Z^ is proper part of Z„ then by (100) m<^n, and since m cannot be =;?, because otherwise Z^ = Z«, we must have m<^nj which was to be proved. 107. Theorem. Z„ is proper part of Z„.. The proof follows from (106), because by (91) 108. Theorem. Z„,^2n(Z„, n'). Proof. For every number contained in Z^, by (98) is <«', hence either =«' or < n\ and therefore by (98) element of Z„. Therefore certainly Z,^>^')Xi{^Z^^, «'). Since conversely by (107) Z^'^Z^, and by (99) «3Z„,, then by (10) we have m(z„, «')3z,,, whence our theorem follows by (5). 109. Theorem. The transform Z\ of the system Z„ is proper part of the system Zn- Proof. For every number contained in Z\ is the transform m' of a number m contained in ^„, and §ince 78 THE NATURE AND mKn, and hence by (94) m' <n', we have by (98) Z'„^Z„.. Since further the number 1 by (99) is con- tained in Z„y but by (71) is not contained in the trans- form Z\, then is Z'„ proper part of Z„-, which was to be proved. 110. Theorem. Z„- = in(l, Z'„). Proof. Every number of the system Z„. different from 1 by (78) is the transform m' of a number m and this must be <«, and hence by (98) contained in Z„ (because otherwise m^n^ hence by (94) also ni'^n' and consequently by (98) ;;/' would not be contained in Z„.); but from m^Z,^ we have m'iZ'„j and hence certainly Z.,3ITT(l, Z'„). Since conversely by (99) 13Z„, and by (109) Z'„^Z„,, then by (10) we have 211(1, Z'„)3Z„. and hence our theorem follows by (5). 111. Definition. If in a system E of numbers there exists an element g, which is greater than every other number contained in jS", then g is said to be the greatest number of the system E, and by (90) there can evidently be only one such greatest number in E. If a system consists of a single number, then is this num- ber itself the greatest number of the system. 112. Theorem. By (98) n is the greatest number of the system Z„. 113. Theorem. If there exists in ^ a greatest number ^, then \s EiZ^ Proof. For every number contained in ^ is <^, MEANING OF NUMBERS, 79 and hence by (98) contained in Z^, which was to be proved. 114. Theorem. If E is part of a system Z„, or what amounts to the same thing, there exists a num- ber n such that all numbers contained in E are <«, then E possesses a greatest number g. Proof. The system of all numbers / satisfying the condition E 3 Z^ — and by our hypothesis such numbers exist — is a chain (37), because by (107), (7) it follows also that E^Z^,, and hence by (87) =g^, where g signifies the least of these numbers (96), (97). Hence also E ^Z^, therefore by (98) every number con- tained in E is <^, and we have only to show that the number g is itself contained in E. This is immediately obvious if ^=1, for then by (102) Z^^, and consequently also ^ consists of the single number 1. But if^is different from 1 and consequently by (78) the trans- form/' of a number/, then by (108) is E^lUiZ^, g); if therefore g were not contained in E, then would E^Zjr, and there would consequently be among the numbers/ a number /by (91) <Cg, which is contrary to what precedes ; hence g is contained in E, which was to be proved. 115. Definition. If /<;w and m<^n we say the number m lies between I and n (also between n and /). 116. Theorem. There exists no number lying be- tween n and n' . Proof. For as 3Pon as ?n<.n', and hence by (93) 8o THE NA TURK AND m<n, then by (90) we cannot have n<^m, which was to be proved. 117. Theorem. If / is a number in Z", but not the least (96), then there exists in T one and only one next less number j, i. e., a number s such that j<;/, and that there exists in T no number lying between s and /. Similarly, if / is not the greatest number in T (111) there always exists in 7" one and only one next greater number u, i. e., a number u such that t <^Uy and that there exists in 7" no number lying between t and u. At the same time in 7" / is next greater than s and next less than u. Proof. If / is not the least number in 7", then let E be the system of all those numbers of T that are </; then by (98) EM,, and hence by (114) there exists in ^ a greatest number s obviously possessing the properties stated in the theorem, and also it is the only such number. If further / is not the greatest number in 7", then by (96) there certainly exists among all the numbers of T, that are > /, a least number w, which and which alone possesses the properties stated in the theorem. In like manner the correctness of the last part of the theorem is obvious. 118. Theorem. In iVthe number n' is next greater than «, and n next less than «'. The proof follows from (116), (117). MEANING OF NUMBERS, 8i VIII. FINITE AND INFINITE PARTS OF THE NUMBER- SERIES. 119. Theorem. Every system Z„ in (98) is finite. Proof by complete induction (80). p. By (65), (102) the theorem is true for nz=\. (T. If Z„ is finite, then from (108) and (70) it fol- lows that Z„, is also finite, which was to be proved. 120. Theorem. If m, n are different numbers, then are Z^, Z„ dissimilar systems. Proof. By reason of the symmetry we may by (90) assume that nK^n; then by (106) Z„, is proper part of Z„, and since by (119) Z„ is finite, then by (64) Z^ and Z„ cannot be similar, which was to be proved. 121. Theorem. Every part E of the number- series N, which possesses a greatest number (111), is finite. The proof follows from (113), (119), (68). 122. Theorem. Every part 6^ of the number-series N, which possesses no greatest number, is simply in- finite (71). Proof. If u is any number in U, there exists in U by (117) one and only one next greater number than u, which we will denote by \p{u) and regard as trans- form of u. The thus perfectly determined transforma- tion j/f of the system 6^ has obviously the property a. iif{U)^U, i. e., C^is transformed in itself by ^. If further u, v 82 THE NA TURE AND are different numbers in U, then by symmetry we may by (90) assume that u<Cv\ thus by (117) it follows from the definition of ^ that if/{u)<v and v<iif/(_v), and hence by (95) \f/{u) <C^^(^') ; therefore by (90) the transforms \l/(u), yp,{v) are different, i. e., 8. the transformation »/r is similar. Further, if ui denotes the least number (96) of the system Uf then every number u contained in U is >z/i, and since generally u<i\p{u), then by (95) ui<, \lf{u)y and therefore by (90) ui is different from «/'(«), i. e., y. the element ui of 6^ is not contained in if/{l7). Therefore ^(C^) is proper part of ^and hence by (64) C/is an infinite system. If then in agreement with (44) we denote by ^^(F), when F is any part of C/, the chain of F corresponding to the transformation if/, we wish to show finally that In fact, since every such chain i/'o(F) by reason of its definition (44) is a part of the system C/ transformed in itself by \f/, then evidently is ij/oi^i) ^^', conversely it is first of all obvious from (45) that the element ui contained in C/is certainly contained in i^^(«i); but if we assume that there exist elements of [/, that are not contained in \f/o{ui), then must there be among them by (96) a least number w, and since by what precedes this is different from the least number «i of the system 6^, then by (117) must there exist in U jilsQ a number v which is next less than w, whence it MEANING OF NUMBERS. 83 follows at once that w^=<\i{y')\ since therefore v<,w, then must v by reason of the definition of w certainly be contained in j/'^Cz^i); but from this bV (55) it fol- lows that also \p(v), and hence w must be contained in if/^^ui), and since this is contrary to the definition of w, our foregoing hypothesis is inadmissible ; therefore C/iif/^^ui) and hence also U=^\p^{ui), as stated. From a, /3, y, 8 it then follows by (71) that ^is a simply in- finite system set in order by i/^, which was to be proved. 123. Theorem. In consequence of (121), (122) any part T'of the number-series N\s finite or simply infinite, according as a greatest number exists or does not exist in T. IX. DEFINITION OF A TRANSFORMATION OF THE NUMBER-SERIES BY INDUCTION. 124. In what follows we denote numbers by small Italics and retain throughout all symbols of the pre- vious sections VI. to VIII., while O designates an arbitrary system whose elements are not necessarily contained in iV^. 125. Theorem. If there is given an arbitrary (sim- ilar or dissimilar) transformation ^ of a system O in itself, and besides a determinate element m in O, then to every number n corresponds one transformation 1^^ and one only of the associated number^system Z„ explained in (98), which satisfies the conditions:* *For clearness here and in the following theorom(i26) I have especially mentioned condition I., although properly it is ?i ppHsec^uence of \\. ap^ III 84 . THE NATURE AND I. i/..(Z,0^O II. ./r„(l) = o> III. \l/nO') = 0il/„(Oy if ^<«> where the symbol Oij/^ has the meaning given in (25). Proof by complete induction (80). p. The theorem is true for n = l. In this case in- deed by (102) the system Z„ consists of the single number 1, and the transformation i}/, is therefore com- pletely defined by II alone so that I is fulfilled while III drops out entirely. a. If the theorem is true for a number n then we show that it is also true for the following number p = n\ and we begin by proving that there can be only a single corresponding transformation if/y, of the sys- tem Z^. In fact, if a transformation xj/^ satisfies the conditions r. ^^(z^)3o ir. ^,(l)-a> Iir. i{/^{^')^Otl/^{m)f when m<^py then there is also contained in it by (21), because Z„^Z^ (l^'^) a transformation of Z„ which obviously satisfies the same conditions I, II, III as «/'„, and therefore coin- cides throughout with i/^„ ; for all numbers contained in Z„, and hence (98) for all numbers m which are </, i. e., <«, must therefore whence there follows, as a special case, gince further by (105), (108)/ js the only number of MEANING OF NUMBERS. 85 the system Z^ not contained in Z„, and since by III' and {fi) we must also have there follows the correctness of our foregoing state- ment that there can be only one transformation i/^^ of the system Z^ satisfying the conditions I', IF, III', because by the conditions ini) and (/) just derived \\ij, is completely reduced to i/^„. We have next to show conversely that this transformation \\ij, of the system Z^ completely determined by {m) and (/) actually satisfies the conditions I', IT, III'. Obviously I' fol- lows from {ni) and (/) with reference to I, and because ^(0)3fi. Similarly IF follows from {ni) and II, since by (99) the number 1 is contained in Z„. The correct- ness of III' follows first for those numbers m which are <;« from {ni) and III, and for the single number m^=n yet remaining it results from (/) and (n). Thus it is completely established that from the validity of our theorem for the number n always follows its valid- ity for the following number/, which was to be proved. 126. Theorem of the definition by induction. If there is given an arbitrary (similar or dissimilar) trans- formation ^ of a system O in itself, and besides a de- terminate element w in 12, then there exists one and only one transformation x^/ of the number-series iV, which satisfies the conditions I. xl;{N)^£l II. ,/.(l) = 0) 86 THE NA TURE AND III. il/(n') = 0{l/{n), where n represents every num- ber. Proof. Since, if there actually exists such a trans- formation \{/, there is contained in it by (21) a trans- formation \f/„ of the system Z„, which satisfies the con- ditions I, II, III stated in (125), then because there exists one and only one such transformation \}/„ must necessarily Since thus if/ is completely determined it follows also that there can exist only one such transformation if/ (see the closing remark in (130)). That conversely the transformation if/ determined by (n) also satisfiies our conditions I, II, III, follows easily from («) with reference to the properties I, II and (/) shown in (125), which was to be proved. 127. Theorem. Under the hypotheses made in the foregoing theorem, if/(iT")=eif/iT), where T denotes any part of the number-series JV. Proof. For if / denotes every number of the sys- tem T, then if/{T') consists of all elements «/'(/')> and Oif/{T) of all elements Oif/(^f); hence our theorem fol- lows because by III in (126) if/(^') = Oif/(/), 128. Theorem. If we maintain the same hypoth- eses and denote by 0^ the chains (44) which corre- spond to the transformation $ of the system Q in itself, then is MEANING OF NUMBERS. 87 Proof. We show first by complete induction (80) that i. e., that every transform ^\l{n) is also element of ^X(u). In fact, p. this theorem is true for « = 1, because by (126, II) i/A(l) = a), and because by (45) a)3^^(o>). 0-. If the theorem is true for a number «, and hence if;(in)'ie,((o), then by (55) also ^(i/^(«))3^,(a>), i. e., by (126, III) \f/{n')iO„{(o), hence the theorem is true for the following number n', which was to be proved. In order further to show that every element v of the chain ^^(o>) is contained in {f/(JV), therefore that we likewise apply complete induction, i. e., theorem (59) transferred to O and the transformation 6. In fact, p. the element oi = i(/(l), and hence is contained in <T. If V is a common element of the chain 6X^^ and the system «/'(iV), then v = »/'(«), where n denotes a number, and by (126, III) we get 0{v) = Oif/{n) = xl/{n), and therefore ^(v) is contained in ij/(JV), which was to be proved. From the theorems just established, {f/{JV)^$X<t>) and ^,(w)^'A(^)> we get by (5) il/(JV) = 0,((o), which was to be proved. 129. Theorem. Under the same hypotheses we> have generally : 88 THE NATURE AND 'A K)=^. ('/'(«)) • Proof by complete induction (80). For p. By (128) the theorem holds for n^=l, since l^ = iVand i/r(l) = o). a-. If the theorem is true for a number n, then ^(^(«,))=^(^X'A(«))); since by (127), (75) ^('/'(«.)) = «AW, and by (57), (126, III) we get ^(;^',) = ^X«A («'))' i.e., the theorem is true for the number n' following n, which was to be proved. 130. Remark. Before we pass to the most im- portant applications of the theorem of definition by in- duction proved in (126), (sections X-XIV), it is worth while to call attention to a circumstance by which it is essentially distinguished from the theorem of dem- onstration by induction proved in (80) or rather in (59), (60), however close may seem the relation be- tween the former and the latter. For while the theorem (59) is true quite generally for every chain A^ where A is any part of a system S transformed in itself by any transformation (f> (IV), the case is quite different with the theorem (126), which declares only the exist- ence of a consistent (or one-to-one) transformation ij/ of the simply infinite system 1^. If in the latter the- orem (still maintaining the hypotheses regarding Q and 6) we replace the number-series 1^ by an arbitrary MEANING OF NUMBERS. 89 chain A^ out of such a system S, and define a trans- formation \\f of A^ in O in a manner analogous to that in (126, II, III) by assuming that p. to every element a o{ A there is to correspond a determinate element \\i{^a) selected from O, and 0-. for every element n contained in A„ and its transform n' = <f>(n), the condition if/(n') = Oil/(^n) is to hold, then would the case very frequently occur that such a transformation ij/ does not exist, since these con- ditions p, a- may prove incompatible, even though the freedom of choice contained in p be restricted at the outset to conform to the condition a-. An example will be sufficient to convince one of this. If the system S consisting of the different elements a and ^ is so trans- formed in itself by <^ that ^' = ^, b' =^a, then obviously a^r=d^^=S; suppose further the system O consisting of the different elements a, yS and y be so transformed in itself by $ that 0(a) = ^, ^(j8) = y, 0{y) = a', if we now demand a transformation {(/ of a^ in O such that if/(^a)^a, and that besides for every element n con- tained in a^ always \lr{n')z=6xp{n), we meet a contra- diction; since for n = a, we get if/^b) =^0{a)=/3, and hence for n = d, we must have ij/(a) = 0{fi)=y, while we had assumed {J/{a)=a. But if there exists a transformation if/ oi A„ in O, which satisfies the foregoing conditions p, <t without contradiction, then from (60) it follows easily that it is completely determined ; for if the transformation x satisfies the same conditions, then we have, generally, 90 THE NATURE AND ^(n) = \l/(^n)f since by p this theorem is true for all ele- ments n = a contained in A^ and since if it is true for an element n of A^ it must by a be true also for its transform n\ 131. In order to bring out clearly the import of our theorem (126), we will here insert a consideration which is useful for other investigations also, e. g., for the so-called group-theory. We consider a system O, whose elements allow a certain combination such that from an element v by the effect of an element to, there always results again a determinate element of the same system Q, which may be denoted by w.v or wv, and in general is to be dis- tinguished from v(o. We can also consider this in such a way that to every determinate element w, there corresponds a determinate transformation of the sys- tem n in itself (to be denoted by w), in so far as every element v furnishes the determinate transform a>(v) = o>v. If to this system n and its element w we apply theorem (126), designating by w the transformation there denoted by 6, then there corresponds to every number n a determinate element \f/{n) contained in Q, which may now be denoted by the symbol w" and some- times called the nth power of a>; this notion is com- pletely defined by the conditions imposed upon it II. u,i = o> III. <o"' = tO 0)", and its existence is established by the proof of the- orem (126). MEANING OF NUMBERS. gj If the foregoing combination of the elements is further so qualified that for arbitrary elements /a, v, CO, we always have o>(i//x) =o)v(/>t,), then are true also the theorems a)"' = a>"co, o>"' w" = (o^^ o)'", whose proofs can easily be effected by complete in- duction and may be left to the reader. The foregoing general consideration may be im- mediately applied to the following example. If S is a system of arbitrary elements, and O the associated system whose elements are all the transformations v of S in itself (36), then by (25) can these elements be con- tinually compounded, since v(6')3^, and the transfor- mation (i)v compounded out of such transformations v and o) is itself again an element of O. Then are also all elements w" transformations of ^ in itself, and we say they arise by repetition of the transformation w. We will now call attention to a simple connection ex- isting between this notion and the notion of the chain to>^(y4) defined in (44), where A again denotes any part of S. If for brevity we denote by ^„ the transform ft)"(^) produced by the transformation a>", then from III and (25) it follows that <u(^„) = ^„,. Hence it is easily shown by complete induction (80) that all these systems A^ are parts of the chain a)^(^) ; for p. by (50) this statement is true for «==!, and 0-. if it is true for a number «, then from (55) and from ^„, = o)(^„) it follows that it is also true for the following number n' , which was to be proved. Since 92 THE NA TURE AND further by (45) ^^o>,(^), then from (10) it results that the system K compounded out of A and all transforms A^ is part of w^^)- Conversely, since by (23) o>(^) is compounded out of q>(^) = ^i and all systems <o(y^„) = ^„,, therefore by (78) out of all systems ^„, which by (9) are parts of K, then by (10) is <o(X)3X, i. e., ^is a chain (37), and since by (9) A^K, then by (47) it follows also that that i^X^^^K. Therefore a)X^)=^, i. e., the following theorem holds : If w is a transformation of a system S in itself, and A any part of S, then is the chain of A corresponding to the trans- formation o> compounded out of A and all the trans- forms o>''(^) resulting from repetitions of in. We ad- vise the reader with this conception of a chain to re- turn to the earlier theorems (57), (58). X. THE CLASS OF SIMPLY INFINITE SYSTEMS. 132. Theorem. All simply infinite systems are similar to the number-series N and consequently by (33) also to one another. Proof. Let the simply infinite system 12 be set in order (71) by the transformation ^, and let w be the base-element of O thus resulting ; if we again denote by Q^ the chains corresponding to the transformation B (44), then by (71) is the following true: a. ^(0)3 a MEANING OF NUMBERS, 93 y. (0 is not contained in ^(O). 8. The transformation Q is similar. If then }\i denotes the transformation of the number- series iV defined in (126), then from y8 and (128) we get first ,A(iv^)==n, and hence we have only yet to show that i/r is a sim- ilar transformation, i. e., (26) that to different num- bers m, n correspond different transforms ^{m), ij/(n). On account of the symmetry we may by (90) assume that m'^^n, hence m^n\, and the theorem to prove comes to this that «/'(«) is not contained in \l/{n\), and hence by (127) is not contained in 6\J/(n^). This we establish for every number n by complete induction (80). In fact, p. this theorem is true by y for n^l, since i/r (1 ) = w and if/{l,)=:t{;(JV) = n. <T. If the theorem is true for a number n, then is it also true for the following number n'; for if ij/^n'), i. e., 6il/{n), were contained in Oij/(n\), then by 8 and (27), ij/^n) would also be contained in i{/{nj) while our hypothesis states just the opposite ; which was to be proved. 133. Theorem. Every system which is similar to a simply infinite system and therefore by (132), (33) to the number-series iVis simply infinite. Proof. If 12 is a system similar to the number- series iV, then by (32) there exists a similar transfor- mation \(/ of JV such that 94 THE NA TURE AND I. ^^{N^=£i^ then we put II. ^(l)=ra). If we denote, as in (26), by ^ the inverse, likewise similar transformation of O, then to every element v of O there corresponds a determinate number '^{y)=zn, viz., that number whose transform ^{n)^v. Since to this number n there corresponds a determinate fol- lowing number </>(«)=«', and to this again a deter- minate element ^{n") in 12 there belongs to every ele- ment V of the system O a determinate element ^{n') of that system which as transform of v we shall designate by B{y). Thus a transformation ^ of ft in itself is com- pletely determined,* and in order to prove our the- orem we will show that by ^ ft is set in order (71) as a simply infinite system, i. e., that the conditions a, y8, y, 8 stated in the proof of. (132) are all fulfilled. First a is immediately obvious from the definition of B. Since further to every number n corresponds an ele- ment v = <f>{n), for which 0(v)=il/{n'), we have gen- erally, III. if/{n')-^eij/(in), and thence in connection with I, II, a it results that the transformations 6, xj/ fulfill all the conditions of theorem (126); therefore ^ follows from (128) and I. Further by (127) and I ^(iV")=.^,/r(iV)=-^(fi), and thence in combination with II and the similarity ♦ Evidently 6 is the transformation t^ </> <p compounded by (25) out of ^, <^. \p. MEANING OF NUMBERS, 95 of the transformation i/r follows y, because otherwise ^{V) must be contained in j/'(^')j hence by (27) the number 1 in N' , which by (71, y) is not the case. If finally /a, v denote elements of O and m, n the corre- sponding numbers whose transforms are \l/(m) = fjL, ij/(n)^v, then from the hypothesis 6(fji) = 6(v) it fol- lows by the foregoing that if/{m') = il/{n')y thence on account of the similarity of j/^, <}> that m' = n', m = n, therefore also /u. = v; hence also 8 is true, which was to be proved. 134. Remark. By the two preceding theorems (132), (133) all simply infinite systems form a class in the sense of (34). At the same time, with reference to (71), (73) it is clear that every theorem regarding numbers, i. e., regarding the elements n of the simply infinite system TV^set in order by the transformation <^' and indeed every theorem in which we leave entirely out of consideration the special character of the ele- ments n and discuss only such notions as arise from the arrangement <f>, possesses perfectly general validity for every other simply infinite system CI set in order by a transformation d and its elements v, and that the passage from iVto 12 (e. g., also the translation of an arithmetic theorem from one language into another) is effected by the transformation ip considered in (132), (133), which changes every element n of iVinto an element v of O, i. e., into xj/in). This element v can be called the nth element of O and accordingly the number n is itself the nth. number of the number- 96 THE NATURE AND series N. The same significance which the transfor- mation <^ possesses for the laws in the domain iV, in so far as every element n is followed by a determinate element <f>{n) = n', is found, after the change effected by if/, to belong to the transformation 8 for the same laws in the domain O, insofar as the element v^^i/^C^) arising from the change of n is followed by the ele- ment 8{y') = \p{n') arising from the change of n'; we are therefore justified in saying that by i/^ <^ is changed into 6, which is symbolically expressed by d = il/<f>{j/f <li = i}/6i(/. By these remarks, as I believe, the defini- tion of the notion of numbers given in (73) is fully justified. We now proceed to further applications of theorem (126). XI. ADDITION OF NUMBERS. 135. Definition. It is natural to apply the defini- tion set forth in theorem (126) of a transformation «/^ of tiie number-series iV, or of the function xj/^n) deter- mined by it to the case, where the system there de- noted by O in which the transform ij/i^) is to be con- tained, is the number-series iV itself, because for this system O a transformation ^ of 12 in itself already ex- ists, viz., that transformation <^ by which iVis set in order as a simply infinite system (71), (73). Then is alsofi = iV, 6 (n) =: <f> {n) = n' , hence I. ^(A^)-37V, and it remains in order to determine xj/ completely MEANING OF NUMBERS. 97 only to select the element w from O, i. e., from N, at pleasure. If we take (o = l, then evidently \\i becomes the identical transformation (21) of N, because the conditions ,/.(l) = l, ,/r(;/) = (^(«)y are generally satisfied by y^{ii)z=n. If then we are to produce another transformation i/r of N, then for w we must select a number m' different from 1, by (78) con- tained in N, where m itself denotes any number ; since the transformation \\i is obviously dependent upon the choice of this number m, we denote the correspond- ing transform ^{n) of an arbitrary number ;? by the symbol m-\-n, and call this number the sum which arises from the number m by the addition of the num- ber «, or in short the sum of the numbers m, n. Therefore by (126) this sum is completely determined by the conditions* II. m^l=m', III. m -\- n' =z (m -\- fi)' . 136. Theorem, m' -\- n = m-\-n'. Proof by complete induction (80). For p. the theorem is true for n=^l, since by (135, II) andby (135, III) (;/^+ly = ^^ + l'• • *The above definition of addition based immediately upon theorem (126) seems to me to be the simplest. By the aid of the notion developed in (131) we can, however, define the sum m+n by (^«(w) or also by <})^(n), where <f> has again the foregoing meaning. In order to show the complete agreement of these definitions with the foregoing, we need by (126) only to show that if <^«(;«) or c^w^Cw) is denoted by \p{n), the conditions \(/{i)=m', \lt{n')=<t>\(,{n) are fulfilled which is easily done with the aid of complete induction (80) by the help of (131). 98 THE NATURE AND <r. If the theorem is true for a number «, and we put the following number «'=/, then is m' -\- n^n m -\- py hence also {ni' -\- ti)' = {m -\- py , whence by (1 35, III) m' -\-p^=m-\-p'\ therefore the theorem is true also for the following number /, which was to be proved. 137. Theorem, m' -\- n = {m -j- «)'. The proof follows from (136) and (135, III). 138. Theorem. l-\-n = n'. Proof by complete induction (80). For p. by (135, II) the theorem is true for n = \. a-. If the theorem is true for a number n and we put n' =p, then 1 -f « =/> therefore also (1 -f n)'=p\ whence by (135, III) l-\-p^/, i. e., the theorem is true also for the following number/, which was to be proved. 139. Theorem. ■i^n = n-\-l. The proof follows from (138) and (135, II). 140. Theorem. m-\-n = n-]-m. Proof by complete induction (80). For p. by (139) the theorem is true for n = l. a. If the theorem is true for a number n, then there follows also (m -\- n)' = (n -\- m)' , i. e., by (135, III) m -f «'= n + m', hence by (136) m -\- «'= n' -]-m; there- fore the theorem is also true for the following number «', which was to be proved. 141. Theorem. (/-\-m)-{- n = /-\-(^ni-\-n). Proof by complete induction (80). For MEANING OF NUMBERS. 99 p. the theorem is true for.fz = l, because by (135, II, III, II) (/+ m) + 1 = (/+ ;;/)'=/+ w'=/+ (w+1). a-. If the theorem is true for a number n, then there follows also {{I -\- ^n) -\- ny := {I -\- {m -\- n)y J i.e., by (135, III) (/+ ;//) -\-n' = l^ {in + «)'===/-]- {m + «'), therefore the theorem is also true for the following number n\ which was to be proved. 142. Theorem. mA^n'^m. Proof by complete induction (80), For p. by (135, II) and (91) the theorem is true for n = \. a. If the theorem is true for a number n, then by (95) it is also true for the following number n', be- cause by (135, III) and (91) m -{- n = {m -\- ny "^ m -\- n, which was to be proved. 143. Theorem. The conditions w >« and w -|- ;2> a-{-n are equivalent. Proof by complete induction (80). For p. by (135, II) and (94) the theorem is true for n = l. (T. If the theorem is true for a number n, then is it also true for the following number n, since by (94) the condition m-\- n'> a-\- ?i is equivalent to (w + «)'> (^a-\-ny, hence by (135, III) also equivalent to w + ;/>«+ «', which was to be proved. loo THE NATURE AND 144. Theorem. \im^a and «>/^, then is also ni -\- ?i^ a -\- b . Proof. For from our hypotheses we have by (143) tn -\- n^ a -\- n and n-\- a^ b-\- a or, what by (140) is the same, a-\-n^a-\- b^ whence the theorem follows by (95). 145. Theorem. \i m-\-n=^a-\-n, Xh^n vi^a. Proof. For if m does not =«, hence by (90) either tn^a or m<^a^ then by (143) respectively /«-|-«> a-\-n or m-\-n<^a-\^n, therefore by (90) we surely cannot have w-j-« = «-|-«, which was to be proved. 146. Theorem. If /> «, then there exists one and by (157) only one number m which satisfies the con- dition fn-\- n = l. Proof by complete induction (80) . For p. the theorem is true for n=il. In fact, if />1, i. e., (89) if / is contained in iV', and hence is the transform m' of a number m, then by (135, II) it fol- lows that l=zm-\- 1, which was to be proved. <r. If the theorem is true for a number n, then we show that it is also true for the following number «'. In fact, if /> «', then by (91), (95) also /> n, and hence there exists a number k which satisfies the condition l=k-\- n ; since by (138) this is different from 1 (other- wise /would be =«') then by (78) is it the transform m' oi a number m^ consequently l^=m'-\-n, therefore also by (136) l=vi-\- n\ which was to be proved. MEANING OF NUMBERS. loi XII. MULTIPLICATION OF NUMBERS: 147. Definition. After having found in XI an in- finite system of new transformations of the number- series N'm. itself, we can by (126) use each of these in order to produce new transformations \\i of N. When we take O = iV, and d {n) =im-\- n=n-\-m, where w is a determinate number, we certainly again have I. xl;{N)^N, and it remains, to determine i^ completely only to se- lect the element <o from iV at pleasure. The simplest case occurs when we bring this choice into a certain agreement with the choice of 6, by putting w = m. Since the thus perfectly determinate ^ depends upon this number m, we designate the corresponding trans- form \p{n') of any number n by the symbol my^n or 7ti.n or mn, and call this number the />r<?^2^^/ arising from the number m by multiplication by the number n, or in short the product of the numbers w, n. This therefore by (126) is completely determined by the conditions II. m.l=:m III. mn' = mn-\-mj 148. Theorem, m' n = mn-\-n. Proof by complete induction (80). For p. by (147, II) and (135, II) the theorem is true for n = \. 102 THE NA TURE AND <r. If the theorem is true for a number n, we have m' n -\- m' =: {m n -\- Ti) -\- m' and consequently by (147, III), (141), (140), (136), (141), (147, III) m* n' =^mn-\-{n -j-m') = mn-\- (w'-f- n)=mn-{- {tn-\-n') = {m n -\- ni) -\- n' = m n' -\- n'l therefore the theorem is true for the following num- ber n', which was to be proved. 149. Theorem. l.n = n. Proof by complete induction (80). For p. by (147, II) the theorem is true for n = \. a-. If the theorem is true for a number n, then we have l.n-\-l^n-]-l, i. e., by (147, III), (135, II) 1 .n' = n', therefore the theorem also holds for the fol- lowing number n', which was to be proved. 1 50. Theorem. mn = nm. Proof by complete induction (80). For p by (147, II), (149) the theorem is true for n = l. <r. If the theorem is true for a number n, then we have mn-\-m = nm-\-mf i. e., by (147, III), (148) mn=n'm, therefore the the- orem is also true for the following number n\ which was to be proved. 151. Theorem. l{m-\-n) = im-\- In. Proof by complete induction (80). For p, by (135, II), (147, III), (147, II) the theorem is true for « = 1. MEANING OF NUMBERS. 103 0-. If the theorem is true for a number «, we have l{ni ^n)^l={lm^ln)^l', but by (147, III), (135, III) we'have /(w + «) + /= l{m + n)' = l(m + «'), and by (141), (147, III) {Im J^ln)^l=lm-\- {In -\- I) = i m -{- ln\ consequently l{m-\- n') = lin-{- In', i. e., the theorem is true also for the following number «', which was to be proved. 152. Theorem. {m-\- n)l=zml-\-nL The proof follows from (151), (150). 153. Theorem. (lm)n = l{nin). Proof by complete induction (80). For p. by (147, II) the theorem is true for n = \, <T. If the theorem is true for a number n, then we have (lm)n-{- /m==/{mn) -\- /m, i. e., by (147, III), (151), (147, III) {lni')n' ^l{nin-\- m^^i^Kjun'), hence the theorem is also true for the following num- ber n', which was to be proved. 154. Remark. If in (147) we had assunied no re- lation between w and 6, but had put (o = /^, 6{n) = m -|- «, then by (126) we should have had a less simple transformation \j/ of the number-series N; for the num- ber 1 would 1/^(1) = /^ and for every other number (therefore contained in the form n'^ would j^(«') = mn-\-k; since thus would be fulfilled, as one could 104 THE NATURE AND easily convince himself by the aid of the foregoing theorems, the condition if/(^n')=6ij/{n), i. e.^ \l/{n') = m-\-\}/{n') for all numbers n. XIIL INVOLUTION OF NUMBERS. 155. Definition. If in theorem (126) we again put n = i\^, and further <i} = a, 6(n)=an = na, we get a transformation ^ of iV which still satisfies the condi- tion I. xf;(iJV)^J\r; the corresponding transform xf/(n) of any number n we denote by the symbol a", and call this number a power of the base a, vifhile n is called the exponent of this power of a. JHence this notion is completely de- termined by the conditions II. a^=<i III. a*"' ^=a.a*' = a*',ao 156. Theorem. ^'«+" = «'« . ^«. Proof by complete induction (80). For p. by (135, II), (155, III), (155, II) the theorem is true for n = \. cr. If the theorem is true for a number n, we have a'~+''.d! = («'".«'•)«; but by (155, III), (135,111) a^+^ ,a = a^*-+*^'' ^oT'^*'', and by (153), (155,111) {cT .a'*)a = ar{a'' .a) = a'" .a*"'-. hence a^-^"' = a'" .a^'y i. e., the theorem is also true for the following number n\ which was to be proved. 157. Theorem. {ary = a'"''. MEANING OF NUMBERS 105 Proof by complete induction (80). For p. by (155, II), (147, II) the theorem is true for « = 1. a-. If the theorem is true for a number «, we have {ary .a"'=::a"''' .or but by (155, III) {ary .a:'' ^iary , and by (156), (147, III) ar''.a'" = a'"''-^'^=^ar''' ] hence {crY=ar''\ i. e., the theorem is also true for the following number n\ which was to be proved. 158. Theorem. {aby = a^.b". Proof by complete induction (80). For p. by (155, II) the theorem is true for n = \. a. If the theorem is true for a number n^ then by (150), (153), (155, III) we have also {aby.a=: .dt(«''./^'')=:(«.«")^'' = ««'.^% and thus {{^al}y.d)l? = {ar'.b''')b\ but by (153), (155, III) {{^aby.d)b = {ahy. {ab')=(^aby\ and likewise {a"' .b")b = a""'. (/^«. b) = a"'. ^«'; tnerefore {aby' = a"\b"', i. e., the theorem is also true for the following number n\ which was to be proved. XIV. NUMBER OF THE ELEMENTS OF A FINITE SYSTEM. 159. Theorem. If % is an infinite system, then is every one of the number-systems Z„ defined in (98) similarly transformable in 2 (i. e. , similar to a part of %)y and conversely. Proof. If % is infinite, then by (72) there certainly exists a part T of % which is simply infinite, there- io6 THE NATURE AND fore by (132) similar to the number-series N^ and con- sequently by (35) every system Z„ as part of N is sim- ilar to a part of 7", therefore also to a part of S, which was to be proved. The proof of the converse — however obvious it may appear — is more complicated. If every system Z„ is similarly transformable in % then to every number n corresponds such a similar transformation a„ of Z^ that a„(Z„)3:S. From the existence of such a series of transformations a„, regarded as given, but respect- ing which nothing further is assumed, we derive first by the aid of theorem (126) the existence of a new series of such transformations »/r„ possessing the spe- cial property that whenever w<«, hence by (100) Z^^Z„y the transformation j/^,„ of the part Z„^ is con- tained in the transformation «^„ of Z^ (21), i. e., the transformations \\i^ and »/r„ completely coincide with each other for all numbers contained in Z^, hence al- ways In order to apply the theorem stated to gain this end we understand by O that system whose elements are all possible similar transformations of all systems Z„ in % and by aid of the given elements a„, likewise contained in fi, we define in the following manner a transformation ^ of fi in itself. If ^ is any element of fi, thus, e. g., a similar transformation of the de- terminate system Z„ in 2, then the system a„.{Z„i) cannot be part oi P{Z„), for otherwise Z„, would be MEANING OF NUMBERS. 107 similar by (35) to a part of Z„, hence by (107) to a proper part of itself, and consequently infinite, which would contradict theorem (119); therefore there cer- tainly exists in Z„, one number or several numbers p such that (x„\f) is not contained in ^(Z„) ; from these numbers p we select — simply to lay down something determinate — always the least k (96) and, since Z„, by (108) is compounded out of Z„ and n' , define a trans- formation y of Z„, such that for all numbers m con- tained in Z„ the transform y{ni)=zp{ni) and besides y(;z')=a„-(/'); this obviously similar transformation y of Z„, in S we consider then as a transform 6{^) of the transformation y3, and thus a transformation 6 of the system O in itself is completely defined. After the things named O and in (126) are determined we se- lect finally for the element of O denoted by w the given transformational; thus by (126) there is determined a transformation xp of the number-series N'ln fl, which, if we denote the transform belonging to an arbitrary number n, not by \j/(n) but by \p^, satisfies the condi- tions II. «/ri=ai III. ,/.,, = ^(,/.„). By complete induction (80) it results first that xp^ is a similar transformation of Z^ in :§ ; for p. by II this is true for n = l. a. if this statement is true for a number n, it fol- lows from III and from the character of the above de- scribed transition 6 from /8 to y, that the statement is io8 THE NA TURK AND also true for the following number n', which was to be proved. Afterward we show likewise by complete in- duction (80) that if m is any number the above stated property actually belongs to all numbers «, which are >/«, and therefore by (93), (74) belong to the chain m„\ in fact, p. this is immediately evident for n = m, and <T. if this property belongs to a number n it follows again from III and the nature of 6, that it also belongs to the number «', which was to be proved. After this special property of our new series of transformations «/r„ has been established, we can easily prove our the- orem. We define a transformation ^ of the number- series Nj in which to every number n we let the trans- form ^(n) = if/„{n) correspond; obviously by (21) all transformations if/„ are contained in this one trans- formation X- Since ij/^ was a transformation of Z„ in S, it follows first that the number- series JVis likewise transformed by x in X hence x(^) ^S- If further m, n are different numbers we may by reason of sym- metry according to (90) suppose m<^n\ then by the foregoing xi.^i) = ^^{m) = xli„{m), and x(«) = '/'«(«) ; but since ^^ was a similar transformation of Z„ in % and m, n are different elements' Ot Z„, then is ^„{m) different from «/'„(«), hence also x{m) different from X(«), i. e., X is a similar transformation of N. Since further Wis an infinite system (71), the same thing MEANING OF NUMBERS. 109 is true by (67) of the system x(^) similar to it and by (68), because x(^) is part of 5, also of % which was to be proved. 160. Theorem. A system 2 is finite or infinite, according as there does or does not exist a system Z„ similar to it. Proof. \{'% is finite, then by (159) there exist systems Z„ which are not similarly transformable in 2; since by (102) the system Z\ consists of the single number 1, and hence is similarly transformable in every system, then must the least number k (96) to which a system Z^ not similarly transformable in ^ cor- responds be different from 1 and hence by (78) ==«', and since n <^n' (91) there exists a similar transforma- tion \p of Z„ in S ; if then j/^ (Z„) were only a proper part of % i. e., if there existed an element a in ^ not con- tained in i/r(Z„), then since Z,,, = Vri(^Z^, «') (108) we could extend this transformation i/r to a similar transformation \p of Z„, in 5 by putting ^{n') =a, while by our hypothesis Z„, is not similarly transformable in S. Hence \f/(Z„)='%, i. e., Z„ and S are similar systems. Conversely, if a system S is similar to a system Z„, then by (119), (67) :S is finite, which was to be proved. 161. Definition. If S is a finite system, then by (160) there exists one and 'by (120), (33) only one single number n to which a system Z„ similar to the_ system 5 corresponds ; this number n is called the number \Anzahl'\ of the elements contained in 2 (or no THE NATURE AND also the degree of the system 2) and we say 5 consists of or is a system of n elements, or the number n shows /i^o/ w««y elements are contained in 2.* If numbers are used to express accurately this determinate prop- erty of finite systems they are called cardinal numbers. As soon as a determinate similar transformation i/r of the system Z„ is chosen by reason of which i/^(Z„) = Z, then to every number m contained in Z„ (i. e., every number m which is <«) there corresponds a determi- nate element ^{m) of the system 5, and conversely by (26) to every element of 2 by the inverse trans- formation ^ there corresponds a determinate number m in Z,,. Very often we denote all elements of 2 by a single letter, e. g., a, to which we append the distin- guishing numbers m as indices so that ^{ni) is denoted by a^. We say also that these elements are counted and set in order by j/^ in determinate manner, and call a^ the m\}i\ element of 2 ; if m<^n then a^. is called the element following a^, and a„ is called the last ele- ment. In this counting of the elements therefore the numbers m appear again as ordinal numbers (73). 162. Theorem. All systems similar to a finite sys- tem possess the same number of elements. The proof follows immediately from (33), (161). 163. Theorem. The number of numbers contained in Z„, i. e. , of those numbers which are <«, is n. *For clearness and simplicity in what follows we restrict the notion of the number throughout to finite systems; if then we speak of a number of cer- tain things, it is always understood that the system whose elements these things are is a finite system. MEANING OF NUMBERS. iii Proof. For by (32) Z„ is similar to itself. 164. Theorem. If a system consists of a single element, then is the number of its elements =1, and conversely. The proof follows immediately from (2), (26), (32), (102), (161). 165. Theorem. If T is proper part of a finite sys- tem 2, then is the number of the elements of T^less than that of the elements of % Proof. By (68) 7* is a finite system, therefore similar to a system Z^, where m denotes the number of the elements of T\ if further n is the number of elements of % therefore 5 similar to Z„, then by (35) T is similar to a proper part E of Z„ and by (33) also Z^ and E are similar to each other ; if then we were to have n<m^ hence Z„5Z^, by (7) E would also be proper part of Z^, and consequently Z„^ an infinite system, which contradicts theorem (119); hence by (90), nK^n^ which was to be proved. 166. Theorem. If V^VCi^B, y), where B denotes a system of n elements, and y an element of r not contained in B, then V consists of «' elements. Proof. For if B = if/{Z^), where i}/ denotes a sim- ilar transformation of Z„, thefi by (105), (108) it may be extended to a similar transformation tj/ of Z„-, by putting il/{n') = y, and we get xl/{Z^)=^V, which was to be proved. 167. Theorem. If y is an element pf a system V 112 THE NATURE AND consisting of n elements, then is n the number of all other elements of F. Proof. For if B denotes the aggregate of all ele- ments in r different from y, then is r=^irt(-^, y); if then b is the number of elements of the finite system B, by the foregoing theorem b' is the number of ele- ments of r, therefore =«', whence by (26) we get b = nj which was to be proved. 168. Theorem. If A consists of m elements, and B oi n elements, and A and B have no common ele- ment, then VTi^A, B) consists oim-\-n elements. Proof by complete induction (80). For p. by (166), (164), (135, II) the theorem is true for n = \. a. If the theorem is true for a number n, then is it also true for the following number n'. In fact, if T is a system of n' elements, then by (167) we can put T = 2Xl{B, y) where y denotes an element and B the system of the n other elements of F. If then A is a. system of m elements each of which is not contained in F, therefore also not contained in B, and we put Xfi(Ay B)^=^%, by our hypothesis m-\-n\s the number of elements of 2, and since y is not contained in 2, then by (166) the number of elements contained in 2n(2, y) = (w + «'), therefore by (135, \U)=m-\-n') but since by (15) obviously ZTKS, y) = Vri{A, B, y) = VCi{A, F), then is w + «' the number of the elements of Va^Ay F), which was to be proved. 169. Theorem. If Aj B are finite systems of w, n MEANING OF NUMBERS. 113 elements respectively, then is lXi{,A, B) a finite sys- tem and the number of its elements is <^-|- «. Proof. If B'^A, then m(^A, B)='A, and the number m of the elements of this system is by (142) <w + «, as was stated. But if ^ is not part of ^, and T is the system of all those elements of B that are not contained in A, then by (165) is their number pKn, and since obviously m{A, B) = m{A, T), then by (143) is the number m -\-p of the elements of this system <w-}- ;?, which was to be proved. 170. Theorem. Every system compounded out of a number n of finite systems is finite. Proof by complete induction (80). For p. by (8) the theorem is self-evident for « = 1. 0-. If the theorem is true for a number n, and if 2 is compounded out of ti finite systems, then let A be one of these systems and B the system compounded out of all the rest; since their number by (167) =^n, then by our hypothesis ^ is a finite system. Since obviously 2=:2Tt(^, B^, it follows from this and from (169) that 2 is also a finite system, which was to be proved. 171. Theorem. If «^ is a dissimilar transformation of a finite system ^ oi n elements, then is the number of elements of the transform j/^(5) less than n. Proof. If we select from all those elements of 2 that possess one and the same transform, always one and only one at pleasure, then is the system T of all 114 THE NATURE AND these selected elements obviously a proper part of 2, because i/r is a dissimilar transformation of % (26). At the same time it is clear that the transformation by (21) contained in «/r of this part 7" is a similar trans- formation, and that ^(^T^ =»^(S); hence the system i/'CS) is similar to the proper part 7" of 2, and conse- quently our theorem follows by (162), (165). 172. Final remark. Although it has just been shown that the number m of the elements of i/'(2) is less than the number n of the elements of % yet in many cases we like to say that the number of ele- ments of j/^(:S) = «. The word number is then, of course, used in a different sense from that used hitherto (161); for if a is an element of S and a the number of all those elements of S, that possess one and the same transform j/^(a) then is the latter as ele- ment of «/'(2) frequently regarded still as representa- tive of a elements, which at least from their deriva- tion may be considered as different from one another, and accordingly counted as ^-fold element of j/'(1S). In this way we reach the notion, very useful in many cases, of systems in which every element is endowed with a certain frequency-number which indicates how often it is to be reckoned as element of the system. 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